Day 126 – Practicing Math Live – Ch. 1 Limits

[00:00:00]Hey, hey. What's up, YouTube? How's it going? Another day practicing calculus.

[00:00:06]Today, I think that we're going to spend the entire stream on section three on delta epsilon proofs, which uh three Mizzy had a funny comment that that's rage bait cuz everybody in the world calls them epsilon delta proofs. I don't know why Tomolman chose delta epsilon.

[00:00:26]It's very strange. I think it's rage bait to get people to check out this book cuz it's a wonderful book and it's hard to get attention um you know when you're publishing a new calculus book.

[00:00:41]All right, so I am joined by my beautiful kitty cat Luna. She's right over here right off camera. I expect that she's going to be really cute today. I hope so. What's up? How's it going? Geometry. What up, dude? Okay, so let's get back to it. I So, this is the farewell calculus tour that we're currently on. Farewell calculus for right now. Now is not the time of my life where I'm it seems like I should relearn all of calculus.

[00:01:09]Um, and that's okay. That's totally okay if I get because right now I want I want onethird of my time of these streams to be um practicing math that's going to make me a better school teacher and then twothirds of the time is math that is more for me like a little bit more for me as a hobby but also some teaching practice. So, it still benefits me as a school teacher. But, uh, yeah, we've got high school, college, and graduate level math. Um, so right now, this is the end of the week on college level math with calculus. Next week, I don't know what's going to happen, but we're going to switch over to graduate level math. And I don't know what that's going to be like. I'm really curious cuz there's a lot of new people in the chat who are probably going to be confused and that's okay. All right. So, yesterday we did a fair number of problems um in Tolman. She has a really excellent uh organization I think in this book. Uh so this is section 1.3.

[00:02:31]um delta epsilon proofs the bookmarks is optional but I just I want to make sure everybody sees these everybody who's here in my stream because these are very important this is conceptual pure math calculus like begins with these proofs um so the book is very well organized in these little um blocks. We've got thinking back, uh concepts, skills, applications, proofs, and then thinking forward.

[00:03:18]I believe that's all of the sections.

[00:03:20]Let me confirm that.

[00:03:25]Yeah, thinking back, concepts, skills, applications, proofs, and thinking forward. And what's cool is we have now hit one of each of those different types of problems. So, the first problem that we're going to work is a thinking forward problem.

[00:03:47]So, uh, Tolman 1.3 point, uh, well, there's not a number for it, but 1.3 thinking forward number one.

[00:04:20]Uh, let's copy this. Hey, hey, bro.

[00:04:23]Cidris, thank you, dude. Um, yeah. Yeah.

[00:04:27]Yeah. This is now going to be a new thing because I don't really want the old live streams to be visible to new people. Like, these streams are not designed to be rewatched. Like if you're enjoying them and that's why you signed up, I'm going to take note of that because like the live stream VODs are confusing a lot of people about like what should I be watching? Should I be watching the live streams? Should I be watching something else? Bro, thank you so much. Cidris, is there a certain type of problem or like a topic you'd like me to to work a problem from today? Thank you, bro. Uh, Cyrus 2856, let's go, dude. Thank you for uh becoming a channel member, dude. Um, can I What's the level of limits you're going to explain today? We are going to be working a delta or an epsilon delta.

[00:05:30]Tolman has gotten it in my in my mouth and that's terrible. These are not delta epsilon proofs. These are epsilon delta proofs like anyone else in the world ever calls them epsilon delta proofs.

[00:05:46]Um very simple ones but still. Ah you want to see some linear algebra? I'll do a linear algebra problem for you bro.

[00:05:53]Thank you. Um, what is there a topic in linear algebra uh that you'd like to see?

[00:06:01]I could I mean, here are the chapters from the book that I'm working. Um, I'd be happy to to grab one. Uh, sorry. Oh, can a Oh, wait. A 12th grader. Yes.

[00:06:13]Yeah. A 12year-old cannot. I'm not like I do not want 12-year-olds in the chat.

[00:06:19]There's certain laws and things and different countries have different regulations. Not dealing with that. 13 plus in the in the live streams. But yes, a 12th grader. Yeah, that's like an 18year-old, right? Um uh you're now watching my past linear algebra live streams. Okay. Okay. Using frollay. Heck yeah. Yeah. Yeah. Of course an 18-year-old can. Yeah. Any I think anybody can watch. Now, you may not understand everything. Next week, I'm going to be doing graduate level math that you probably won't understand, and that's okay. Um, okay. Well, if you want to see like a random topic, Cidris, if it's okay with you, I'd love to to show you a vector space problem. Have you seen vector spaces yet? Um, vector spaces is really to me the heart of linear algebra and this is what turned me on to becoming a math major.

[00:07:21]Um, okay bro, thank you so much.

[00:07:27]Let's work. Have you seen vector spaces yet? I will go off topic. Like the stream topic today is calculus, but if somebody signs up as a channel member, I will work any topic you want to see.

[00:07:41]Thank you. I really appreciate it. Um, if you haven't seen this yet, I can Ooh, you're only on the first chapter. Okay.

[00:07:51]Ooh, where what do I show you? I would like to still show you vector spaces.

[00:07:58]Okay. um if that's okay with you like it's a little more abstract than uh than the first chapter but this is where the where linear algebra starts becoming really beautiful and I can connect it to the first chapter like I will try to explain vector spaces to you um assuming that you haven't seen them before uh you will try your best calculus is fun doesn't matter if it's graduate level well next week like actually today We're today might be the last day we do calculus. So, I know that this is very confusing. Um, you know, I promise there is a structure. There is a method to the madness. Um, and we'll we'll understand it as we go. Okay. Heck yeah. Yeah, you're okay with that? Unless you wanted to go through limits today. I'm going to pause. If you want to see limits instead, I'll do it. But I would rather show you some math that you want to see.

[00:08:55]Um, if you're working linear algebra, I'd love to work a linear algebra problem. Um, and bro, you are going I'm you know what I want I want to show you some linear algebra. Uh, dude, thank you so much. Why isn't my pen working? Cidris 2856.

[00:09:24]You are now permanently.

[00:09:27]Look at this.

[00:09:30]You're right in there. You're right in there with this linear algebra problem set. Thank you, bro. Let's do a little detour into vector spaces. Um, I think it's nice to Well, yeah, we're we are today working calculus, but let's take a detour.

[00:09:54]Thanks to Cidus uh for getting a new channel membership.

[00:10:08]Uh let's see. So I want to explain a little bit about the concept of a vector space.

[00:10:16]This is the heart of linear algebra. This is really where linear algebra becomes its own field. Uh it a vector space is any um any set uh V with two operations plus and multiplication.

[00:11:03]any set V and a field F where addition will take two vectors and give you a third vector. It'll give you a new vector. So if you have two vectors, you can add them together. And we usually write V1 + V2 equals V3. This is like the standard way, but but there's other there's there's other cool weird scenarios. And then multiplication will take a uh a field element and a vector and then map to a scalar multiple of that vector. So this is like in um so for example in vectors in like Rn right you can multiply a vector by a number right 2 * the vector 3 4 is 68 right so here your k is 2 and your vector is 34, right? Uh and we get the same thing with matrices or single numbers. Um in this book, usually V and K usually is Rn and R like well this is vectors. Um really hm let me fix that.

[00:13:04]Most often in this book we are working over n by m matrices with real numbers.

[00:13:14]I'm sorry if this is like a little overkill right now but I I think that you might be able to be following me here. So, matrices and scalers.

[00:13:30]Let me know if I'm if I've totally lost you. Let me know if I've totally lost you. If so, I will What will be the duration today? I'm hanging out for one, two, maybe three hours. Holy Hopefully two.

[00:13:46]Uh, what's the point of members only stream when I can just copy the link and download it? Yeah, I mean that's fine.

[00:13:53]You can um it's fine with me, but it's it's partly like memberships is partly to help support what I'm doing if you can afford it. You know, I don't want anybody to be paying anything if if you can't afford it, right? Like I I want to find ways that you all can help me try to make this a sustainable full thing.

[00:14:16]Yeah. Yeah. So these are scalers.

[00:14:18]Exactly. Exactly. But there's other interesting vector spaces. Um like um like your matrices can have complex numbers and we can be working over complex scalers and pretty much everything in linear algebra still works. like matrices with complex numbers and complex scalers. Uh you could also be working over square matrices.

[00:14:53]So not n by m but mn. This symbol this means square matrices.

[00:15:00]Uh we could also be working over invertible matrices.

[00:15:10]So this is invertible.

[00:15:16]Do you know what the word invertible means yet? Cidris, it's okay if not.

[00:15:24]Um if not, then I'm just I'm going to move on. Um in either case, I should probably move on. Let's get back to the definition of of vector space.

[00:15:41]Um the definition of a vector space is a set uh v and um a field. It's important that this one is a field and I don't want to get too far into what that means but field f or k what did I use before? K is more common in algebra. Axel uses F, but I like K. Um, uh, with, uh, welldefined addition and scalar multiplication.

[00:16:20]Okay. Now we need to satisfy eight axioms.

[00:16:25]We have four addition axioms.

[00:16:30]A1, A2, oops, A3, A4.

[00:16:37]And we have four uh scalar multiplication axioms. S1, sub_2, S3 and S4.

[00:16:48]Um, now what are they?

[00:16:54]Both of them need to be associative.

[00:17:01]Okay, what does that mean? Let me show you A1 associative.

[00:17:12]This means that V + W T U VW, let's go. U + V + W. Whatever objects these are, U, V, and W, like they're no longer matrices or vectors or specific things, but anything where our addition can be regrouped in this way that we're used to.

[00:17:41]Um S1 associivity says that um I like K's and L's. K L times a vector has to be equal to K * LV.

[00:18:06]Okay. Uh so on this one the on the on the left hand side your scalar you have one scaler KL multiply by the vector and then on this one uh V maps to LV maps to KLV right it's like two successive um multiplications.

[00:18:39]So this is associivity.

[00:18:43]Okay, nothing too surprising. You could think about how that connects with what you've seen so far. Um on addition, you need to have the existence of I'll write it in fancy symbols. There exists there has to be a zero.

[00:19:04]Um or in other words, additive identity.

[00:19:14]Um a A2 identity.

[00:19:23]What does that mean? That means there exists some zero in the vector space such that uh for all vectors v + 0 is v.

[00:19:49]Okay, so this acts just like what we've seen so far. There's always a zero vector or a zero matrix. Um, okay. What else do we need? We need inverses.

[00:20:08]Um, so additive inverses.

[00:20:17]Um, for all the there is a negative v such that v + negative v equals zero. So we have subtraction quote unquote like there's always an additive inverse. Uh, and so far it's always the same as -1 * the matrix or -1 * a a vector. Um, and then what is the last one? Oh, we need commutativity.

[00:21:00]Commutativity.

[00:21:02]So A4 commutativity we need for all vectors V + W equals W + V.

[00:21:23]So commutativity is another requirement.

[00:21:28]uh for scalers for scalar multiplication we need associivity.

[00:21:34]What else do we need?

[00:21:37]I know S3 and S4 very quickly. What is S2?

[00:21:47]Why am I blinking right now? What is S2?

[00:21:51]I think it's the existence of a a multiplicative identity. It must be.

[00:21:59]Yeah. Yeah. Yeah. Yeah. So again, we have an identity S2 identity.

[00:22:12]I don't really like this color scheme, but whatever. S2 identity means it's a scalar multiple identity.

[00:22:20]uh for all oh well oops there exists some element one in the field and in fact because it's a field there already is there must be there must be a one in the field um And it satisfies for for the for the um multiplicative identity in the field it satisfies 1 * v = v for all vectors.

[00:23:12]Okay. Um and then we have distribution the distributive property with two vectors and the distributive property with two scalers.

[00:23:30]Uh let's get those written down. S3 and S4.

[00:23:37]I'm just going to try to write these more quickly. uh S3 S4 um are that if you have one field element and two vectors that we can combine multiplication and and addition in this way KV + K W right and then if we have K + L * 1 vector then similarly This is K * a vector V plus L * a vector V. So a lot of books will really emphasize that these vectors are different objects, you know, with some kind of like arrow or something. Just remember that these are totally different objects. And so it's not obvious that multiplying with an addition is going to satisfy the same distributive property that the regular numbers have. Um, okay. Invertible means the matrix has an inverse that exists. Yes, exactly. Oh, I forgot to um put chat up on screen. On screen. Oh, why didn't Yeah, that didn't work. That is outdated.

[00:25:04]H.

[00:25:11]Can I refresh?

[00:25:14]There we go.

[00:25:17]Oh, do you all prefer the vibes of the Do you all prefer the Hawaii stuff? Cuz I'm down to just keep the Hawaii stuff if you prefer. Uh, what's up, Tamwa?

[00:25:28]Dude, how are you? Okay, so these are the axioms or the definition of a vector space. Um, let's see. Can I I'm going to work number 16 very quickly.

[00:25:52]Thank you, dude. Uh, Frolley 31 uh 16 asks us if the set this is one of my favorite examples that I think of that's not matrices not uklidian vectors PN is the set of all polomials with degree less than or equal to n.

[00:26:35]Oh, degree less than or equal to n.

[00:26:40]real coefficients um and the zero polomial.

[00:26:55]Yeah, he he says along with the zero polomial but I think zero we typically just consider to be a polomial anyway. So this is now the set of all polomials with degree less than or equal to n uh with real coefficients.

[00:27:16]And the question is is um PN the set of polomials degree up to n um a vector space over the real numbers.

[00:27:40]Okay. The first is addition is addition well defined. Yes.

[00:28:05]If you have any two polomials of degree up to n um oops and we add with another polomial degree up to And if we add those two polomials together then we get another polomial with degree less than or equal to n.

[00:28:59]Cool.

[00:29:07]So now this is again another nth degree polomial.

[00:29:14]Um it's possible that the first that coefficients can be zero. So it's it's very important to say degree up to n. Um for example uh x^2 + 2x + 3 + um first I'm going to give a this kind of example 4x^2 + 5 x + 6 this is these are two second degree polomials and this is 5 x^2 + 7 x + 9 and this is a secondderee polomial, right? Um, alternatively, Luna, Luna, Luna, Luna, you're cute, but you're annoying.

[00:30:08]Also, um, x2 + x2 is 0.

[00:30:19]But this is still degree less than two.

[00:30:24]Like we still say that the that this or um x2 + -x^2 + x or something that is x and it's still degree less than 2.

[00:30:44]So um yes addition is well defined and it's you know if you take two polomials you get another polomial of the degree up to the original degree. Uh then the next question is is um multiplication of a field element and a polomial to a nth degree polomial. Well defined and in in this um example the field is the real numbers.

[00:31:30]Yes. Um, a real number times a polomial is equal to R A N X plus dot dot dot R A1 X plus R A KN.

[00:31:58]Right? And now this is like some new just some new uh coefficients that are all real numbers.

[00:32:12]Uh I think that the book is usually alternating R and S to be real numbers.

[00:32:25]Uh Luna, calm down. calm down.

[00:32:32]Right? So now this is still an nth degree polomial.

[00:32:38]So we have those two properties and now we need to check a1 a2 a3 a4.

[00:32:50]Um A1 again is the do we have associivity?

[00:32:58]Um, yeah, you can convince yourself trying this that P1 + P2 P3 does equal P1 + P2 plus P3.

[00:33:20]And that's because because um of associivity of coefficients like all the coefficients um all the coefficients have additive associivity.

[00:33:41]um because all the you add the coe corresponding coefficients and then those are associative and you get associivity. Um then we need um identities.

[00:34:01]Um well zero is the the additive identity polomial for any polomial the um the zero polomial satisfies this.

[00:34:21]Uh what else do we need? We need inverses.

[00:34:28]Um for any polomial um negative p would be negative coefficients.

[00:34:40]You just take all the coefficients to be negative.

[00:34:45]Um and now this is itself a polomial and if you add that to any other one then there's zero.

[00:35:00]Uh and once more commutativity yes commutativity um yes any two polomials um commute under addition because of um the commutativity of the real numbers.

[00:35:33]Um, we need to check S1, S2, S3, and S4.

[00:35:40]And I will let you treat this as an exercise if you want to to to work this one and see. So, this is a rapidfire intro, I feel, to something that you've probably not seen before, but this is one of the, in my opinion, one of the most beautiful examples of a vector space and it follows like all the same structure um of the matrices that you're learning in chapter 1. it like all the same theory works like in this more abstract different example of a of a polomial space the like it's examples just like this that turned me on to proof-based math and becoming interested in pure math. So it was a joy to to share this one with you Cidus. Thank you for becoming a channel member dude. Um, if I don't know the theory very well, I probably won't solve these kind of problems. Yeah, but this was Don't worry if you didn't completely follow this.

[00:36:52]Um, yes, you need to know the theory, but these kinds of problems will help you learn the theory and we just jumped ahead two chapters. Um, so if if you're enjoying this, dude, this is this is some cool interesting stuff. Like it's a proof type of question. Um, please reply and help you. What's up?

[00:37:17]Purchase a membership from an account, but I forgot that mail and I deleted browsing data. Please help. My parents won't give me another membership. Okay, it's with the same name and you also commented on a video in the number theory playlist. Okay, let's go, dude.

[00:37:33]Um, well, how you forgot that mail and you deleted browsing data. Dang.

[00:37:42]Bro, that sucks. Um, I can see if I can find the membership you're talking about. It's Divium. Well, thank you so much for becoming a channel member, man. Like, it really means so much to me. Um, I appreciate it. Let me just check my members list.

[00:38:08]Where? Yeah, I see you. your Divium P6 your Divi P6U. So, let me put you on the thanks page, man. I really appreciate it. Um, bro, Divium P6U.

[00:38:37]that may help you get into that account.

[00:38:42]I don't know.

[00:38:44]Um, and uh, we just had Happy Abe join earlier. Bro, thank you so much, Happy Abe, as well.

[00:38:55]Uh, Divium, can I work a calculus problem for you?

[00:39:00]The cat is on the table. She will be hilarious. Yeah. Uh, we'll do game theory someday. Yeah, but no promises. I don't know when.

[00:39:12]Uh Whoa, bro. You got his account. Yeah, I found it. It's It's there. I I do see that you have a channel membership. I don't know how to solve this. Like, yeah, Divium should have a green name.

[00:39:24]Everybody should recognize that. Um I don't know. I can't like reset your password for you or anything like that, but you could I might I wonder if I can like cancel like disable your membership and then you can sign up for this one or something. I don't really know.

[00:39:47]Um, okay. I don't have access I can't help you get access to the I I don't have a way that I can give you access to that. I would say you can I mean do you have the email that that's associated to? Um cuz you can use the forgot password.

[00:40:15]I mean, honestly, if you if you signed up for a channel membership on an account that's not yours, like on your parents account or something, I I don't know. You shouldn't you shouldn't do that, I guess.

[00:40:28]Uh, yeah. When will you be live for the next calculus session? We're going to get back to calculus. I don't know. Um, today may be the last calculus stream for a while. um because it looks like I'm not going to be getting, you know, that high school job I was shooting for. So, I want to do abstract algebra because there's a lot of people interested in pure math right now. Um I think learning to write proofs is the focus that I should be starting, you know, and like we should be like zoning in a little more on proofs. Um so yeah, we'll do some logic at some point.

[00:41:15]Um that'll I will certainly introduce logic in um we will be doing logic in in the basic math series. I imagine that it's not next week but the week after. I think it'll be the week after that I want to do the logic um stuff.

[00:41:35]Yeah, Olympiad problem someday. I mean, I can't really help with that right now.

[00:41:41]I'm not ready to work those kinds of problems. Um, but you can ask them in the in the Discord. You can join the Discord and there are definitely people in the Discord interested in working those kinds of problems.

[00:41:56]discord.jpractices.com if anybody wants to join. Okay. Uh, let's get back to calculus, though.

[00:42:25]She calls them delta-epsilon limits. I'm going to call them epsilon delta epsilon delta proofs.

[00:42:36]Okay. Um, this next problem is a thinking forward problem.

[00:42:57]Okay, it says the first one says um we still do not have a way to calculate limits easily. We have to do a long proof using epsilons and deltas.

[00:43:17]Uh, we still do not have a way to calculate um limits easily.

[00:43:42]Uh in the following problems you will develop rules for calculating limits of some very simple functions.

[00:43:53]Okay, you know what? I'm just going to copy this Okay, so we still don't have a way to calculate limits easily, but we can develop some rules for calculating limits of some very simple functions. So part one or part a explain why it makes intuitive sense that limit basically it's saying prove that limit um as x uh yeah limit of x= c as x tends to c for any real number C. So this is part one explain intuitively.

[00:45:22]Um and how could I prove that um intuitively?

[00:45:32]The function f ofx= x uh is smooth and equals c at any x= C more precisely F of C equals C everywhere.

[00:46:09]Then C minus epsilon equal So like a tiny bit to the left like 0.00001 to the left is just 0001 less in the in the y value.

[00:46:47]This may be already kind of jumping to the epsilon delta proof.

[00:46:53]Um, so if I wanted to try to use more intuitive words for somebody who hasn't seen epsilon and delta before.

[00:47:05]So it is um arbitrarily arbitrarily close to C.

[00:47:22]Um for X near C, right? Like the value being arbitrarily close to C, like the exact same xcoordinate would equal the same function value.

[00:47:46]Um, here I've already started essentially part two. I think I've got the idea on the intuitive argument for this one. For part two, um, I think we're going to take Delta to be epsilon. Let epsilon be greater than zero. For any epsilon greater than zero, it can be infinitely just like minuscule like a grain of sand that small, right? Let epsilon be greater than zero.

[00:48:25]Then um we see that delta equal epsilon satisfies the epsilon delta definition.

[00:48:50]Let x be such that um so x is not equal to c.

[00:49:17]um which handles like the left hand uh or like the first less than sign and the absolute value of x - c is less than delta which then therefore x - c is less than epsilon.

[00:49:44]I took um I took delta to equal epsilon.

[00:49:54]And then therefore the function value minus the limit value is less than epsilon for any uh epsilon greater than zero.

[00:50:08]where f ofx is x and l is c. Therefore, q e dq.

[00:50:35]Go. Dude, thank you for becoming a channel member, bro. Thank you. Aba Skyed, let me know how to pronounce your name if I got that wrong, bro. Thank you so much. Um, aba skyed sked, dude. I appreciate it.

[00:51:07]Congo Congo M, the member of the channel.

[00:51:12]Okay, what does that mean? Where's the cat? She's sleeping on my computer. It's warm and she loves that right now.

[00:51:20]Can you teach pure uklitian geometry after this? No.

[00:51:25]No. I I'm not Oh, kayed. Okay. Nice.

[00:51:30]Thank you, Kayed, dude. Thank you. All right. Well, I appreciate it. Can I work a calculus problem for you? Or also, um, Divium, is it okay if I give I'm going to give you credit for this, uh, calculus problem we just worked, bro? Thank you so much.

[00:52:02]Did I put it somewhere else already?

[00:52:11]I'm just going to call you Divium because it sounds like the last letters are randomized, right? And then yeah, for Luna level, I'll I'll be working a problem from the section I'm already working most of the time, unless somebody asks for linear algebra, and then I might do linear algebra just because I want to. So, let's do another calculus problem for Aba Skyed. Aba.

[00:52:36]Abaz Kayed. Right. Okay. Thank you, man.

[00:52:45]Abaz Kayed.

[00:52:54]Or I could just call you Kayed if that's what you meant to say. How's it going, Gerson Franklin? How are you? What's up, everybody? Thank you again y'all for helping support my journey. We got we got these channel members supporting calculus so far. You are permanently in my book.

[00:53:15]Name is too difficult. Barely anyone can spell it. Uh I try my best. I think it's important like to try, you know.

[00:53:23]Um okay, so now I've officially done another epsilon delta proof.

[00:53:37]Okay, the next one's a little trickier. I don't know if I want to do that one yet.

[00:53:46]Yeah, we could do it. We could try it.

[00:53:47]I'm going to give you Let's try to work this next one. Thinking forward number two.

[00:54:11]So we have successfully done this first one. Now explain why it makes intuitive sense that limit is x to so this we want to we want to just snag this if I can again.

[00:54:53]F ofx equals H.

[00:55:05]How do I It has no jumps.

[00:56:27]This is my intuitive argument.

[00:56:34]I think I think we got the intuitive argument on this one.

[00:56:41]Um, no, I'm not going to do a differential equation. We're not working on differential equations.

[00:56:49]Um, formally, let epsilon be greater than zero.

[00:57:02]We aim to find a delta greater than zero such that for all x where x - c for all x not equal to c and within a delta neighborhood a punctured delta interval around C then um the function value is near L and in this problem x^2 - c^2 is less than epsilon H.

[00:58:08]So now we begin some scratch work.

[00:58:16]If x^2 - c ^2 is less than epsilon then the absolute value of x + c * x - c is less than epsilon.

[00:58:35]Um I see yeah I see that difference of squares uh which is usually a pretty good sign. And I see that X minus C is right here, which is something we want to get right there.

[00:58:53]Right. So, I think that we're we're close. I will purchase membership again from this account. Dude, I appreciate it, but you don't have to. Like, can you cancel the other membership? Otherwise, you'd have two recurring payments. Also, anybody who wants to sign up for a membership, um, definitely do it from the website or an Android app. Don't do it from the iPhone app because with the iPhone app, Apple takes an extra cut and it's more expensive, which is a bummer.

[00:59:27]So, okay.

[00:59:31]So, I think that we're we're getting somewhere here.

[00:59:35]Um H how do we do this again?

[01:00:41]I think we need to use this lovely inequality called the Koshi Schwarz inequality that I believe is right.

[01:01:14]Bro, I don't know how I can I will look into this. I I hope that I can cancel your membership on your behalf.

[01:01:26]I don't I've never had to deal with this before.

[01:01:41]Yo, we got another hobbyist. Arlene, are you here?

[01:01:46]Dude, I can't believe you all. Thank you so much.

[01:02:00]Um, okay. I want to make sure I don't forget about this because this is important.

[01:02:11]So, Divian, let me try.

[01:02:19]I'll I'll reach out to customer service if I need to. Yeah, I'll I'll figure it out.

[01:02:29]I will figure something out.

[01:02:53]Hey, no way. Your grade 11 brother a great deal. That's awesome. I'm happy to hear that. I'm happy to hear that your brother is enjoying the the videos. That really means a lot.

[01:03:08]Okay, Divium, I'm trying to make sure I remember to delete this stuff.

[01:03:20]Okay, so let's see.

[01:03:27]I mean, the the problem did say hint assume delta is less than or equal to one. So, let's see. the the problem did give this hint and I want to remember that. I just want to see if I can find that point myself but uh if let's see I do remember that it gave that hint and I I haven't yet assumed the hint. I want to see if I can figure out that hint on on my own. So absolute value of x + c x - c has to be less than or equal to the absolute value of x + c * x - c.

[01:04:23]And if we assumed Here.

[01:06:04]I don't know if that works but yeah let's just try to take the hint. So let epsilon be greater than zero um and assume delta is less than one.

[01:06:33]Let's just see if we can find then um if X is not C and close to um is close to but not equal the C.

[01:07:10]First off, x is not c and um x - c is less than delta which is less than one.

[01:07:32]Consider Then f ofx - l is x2 uh - c^ 2 which is x + c * x - See and um absolute value of x minus c is greater than zero and less than one.

[01:08:40]So because um Oops.

[01:09:21]Yeah. I mean, how do we get our epsilon here? Um, how does this work, and like oh I'm so weak. I'm so rusty on my epsilon delta proofs. Very I feel like I've got something along the right idea by using the Koshy Schwarz inequality.

[01:10:15]An absolute value of x - c is less than one.

[01:10:26]Oh, I feel like I'm so close.

[01:10:38]My chat is behind, isn't it? Why is my chat behind?

[01:10:51]We have absolute value x minus c is less than one. Oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh oh okay okay.

[01:11:02]Yeah. Uh let's see that. So absolute value of x - c is between 0 and one.

[01:11:16]which tells us that um -1 is less than x - c which is less than positive 1.

[01:11:30]Um Is there something like this?

[01:12:24]Wait.

[01:12:48]remove zero. Don't remove absolute zero.

[01:12:51]That's what makes it work. We can add C to all parts of the equality. That's so that's what I did in the second line there, right?

[01:12:59]Um, wait. Don't rem remove zero. I did.

[01:13:12]Didn't I remove zero?

[01:13:19]Well, oh, yeah. Um the zero part of this doesn't doesn't lead to the next line. It it doesn't hurt. It doesn't make it false, but the next line is equivalent to the previous one. Right?

[01:13:43]That bounds x + c.

[01:13:48]If I add If I add 2 C to to the whole thing. Yeah. Okay. So then X + 2 or X.

[01:13:59]Okay.

[01:14:00]Okay.

[01:14:04]Yeah.

[01:14:23]x + c is then so I need to add 2 c to both sides of the inequality, right in order to bound that but or is it Um, yeah. Doesn't this say that x + c is less than um how do these freaking work, man. Um, Yeah, it seems like a triangle in like I was thinking of triangle inequality.

[01:16:02]Absolute value of x minus e be can be as little as you want to and x plus c is bounded in x. So yeah, that's what I was that's what I thought I was showing before over here like X plus like by co Koshi Schwarz. we get that third line in this.

[01:16:39]And then because we have a bound on x minus c, then we have this. And then can I Oh, I can maybe do uh a triangle inequality on this.

[01:17:12]How do we get to epsilon though?

[01:17:18]Like we need to define epsilon with without depending on x.

[01:17:43]I feel like we want to make some kind of assumption that delta squared is less than epsilon squared or something cuz like we we haven't written down delta in terms of epsilon anywhere.

[01:18:10]So that feels a little funny.

[01:18:25]Oh.

[01:18:27]Oh, wait. No, we we want to write.

[01:18:34]Can we just take delta to be epsilon?

[01:18:40]I think we would expect that in this kind of problem.

[01:18:46]Yeah, I I think we would expect delta to be epsilon and for it to be between zero and one Now we have an epsilon in this in this thing.

[01:19:22]Right?

[01:19:25]Because epsilon is equal to delta.

[01:19:33]So that thing is this line is less than delta which is equal to epsilon.

[01:19:52]And I believe that. Oh, and this is less than epsilon because delta is less than one. So I think that we know that the absolute value of x + c needs to be less than one as well like because we're taking because x is in um C minus delta to C union C to C + delta.

[01:20:55]So, um, and this is a subset of C minus1, a proper subset of C minus one to C union C C + one.

[01:21:30]So, I just feel like I'm missing like I think I see why absolute value of x + c must be less than one, which would then complete this. But I'm having a hard time writing that down.

[01:21:59]But I see that it's like in this Delta neighborhood.

[01:22:03]Oh man.

[01:22:10]Yeah. We should get delta in terms of epsilon, right?

[01:22:17]Yeah. Divium. Exactly. So this is the more formal pure math version of the definition of limits. This is like the pure math version. And it's it's harder. Um, so I think I think taking delta to equal epsilon should work for this.

[01:23:08]kind of problem.

[01:23:31]new page.

[01:23:35]Let fome be greater than zero.

[01:23:38]Um and In other words, if we want it to be less than or equal to one, we need it to be the minimum of 1 and epsilon, right?

[01:24:20]So delta is less than or equal to 1 guaranteed which is less than or equal to epsilon.

[01:24:41]Then assume Oh wait. Oh uh x - c is strictly less than delta.

[01:26:03]So then x - c is strictly less. There we go. Okay. I was like something feels very wrong here.

[01:26:11]strictly less than one which is less than or equal to epsilon.

[01:26:23]Uh wait no epsilon can be less than one. What am I talking about? No. Yeah this is false.

[01:26:32]This is false.

[01:26:38]That line is not real.

[01:26:49]But we do know for sure that delta is less than or equal to one.

[01:27:01]Delta is bounded above by one.

[01:27:37]Can we try triangle inequality at the beginning?

[01:27:42]Like it just sounds like a Koshy Schwarz since this is like um a product or like a difference of squares. But I mean we could try this.

[01:28:10]I don't I don't think triangle inequality is the way.

[01:28:20]I don't feel like that's the way.

[01:28:30]This gets us our beautiful XUS C and X plus C by Koshi Schwarz. This is this um x uh x - c is less than one.

[01:29:08]So then this x minus c. So now we know that this is strictly less than it's greater than zero and less than one.

[01:29:22]So we have that Any hints, man? I feel like we're so close, dude. Lockheed, what's up, man?

[01:30:10]epsilon is arbitrarily small and you don't choose it. Well, no, you fix you fix an arbitrarily small epsilon like you let epsilon be greater than zero and then you define delta in terms of epsilon.

[01:30:25]So, so on this I define delta as the minimum of 1 and epsilon. If epsilon we chose to be 10, then delta is one. But if epsilon is less than one, we choose epsilon, right? Um, what's up locked? Uh, lock, we are working this problem. I'm trying to do a delta epsilon for limit of x^2= c^2.

[01:31:03]I can write it here. We are trying to triangle to get this one.

[01:31:12]My analysis is so bad.

[01:31:16]But that's why we're here. That's why we're practicing and I'm I'm trying to get back to it. Thank you, Crimson.

[01:31:22]Crimson Gamer, bro. Thank you so much.

[01:31:25]I'm glad you're liking my videos.

[01:31:39]Human joins math live. Human sees real analysis. Human scared. Right, Penjam?

[01:31:46]That's how I feel, dude. That's how I feel.

[01:31:54]Ask Claude. I'm tempted.

[01:31:58]I'm tempted. Delta Epsilon still trips me up. Me, too. Why would we ever call it Delta Epsilon? It's epsilon delta. My chat keeps falling behind. Oh, Sumi Chararma, did you sign up as a Oh, wait.

[01:32:13]No, never mind. Never mind. Never mind.

[01:32:16]Okay. Okay, dude. I swear we are close.

[01:32:50]I was thinking of trying to do something something like well wait delta being the square root of epsilon.

[01:32:59]Oh, I mean then delta is going to be bigger than epsilon, right? Use triangle and substitute the assumption.

[01:33:13]Yeah, I was thinking that we might we might do a triangle.

[01:33:25]Um, can we do something like x - c + 2 c?

[01:34:08]this. Now we've landed in some weird other No, I don't think that this works, does it?

[01:34:25]I don't think that this gets us anywhere, but we do have x - c is less than one. Strictly less than one.

[01:34:41]We want to get back to our epsilon. I mean like here we do have that that that this is less than epsilon.

[01:35:19]But now we're never we're not going to get to something that's less than epsilon. We're always adding positive amount. So that that gets us stuck. So I I feel using a triangle inequality with by rewriting x + c as x minus c doesn't doesn't work.

[01:35:51]I don't think we can get anything with this can we?

[01:35:57]Um there is like a reverse triangle inequality but I think it reverses in the other direction.

[01:36:06]Um h I know somebody's figured this out. Uh x minus e mod less than delta. Then combine these two steps, bro, and you're done. So, I'm close, right? Like, yeah, I don't understand the Hindi conversation. So, lock, you need to let me know.

[01:36:39]I don't think delta greater than epsilon should break anything.

[01:36:44]Maybe not, but I just don't see why bringing in a square is going to help.

[01:37:15]I just don't follow what you're saying here. X - C. What do you mean? Mod mod less than epsilon / C mod C + 1. Are you saying is less than?

[01:37:31]You know what you should do? Draw a graph. I feel I should draw a graph.

[01:37:43]I don't know what spammers you all are talking about. I'm not seeing Oh, I'm not seeing everything. You all are seeing more than I am. I only see top chat. If I look at live chat, there's like a lot more spam going on.

[01:37:58]Yeah, Lockheed, if there's spam that you're saying, please feel free to time out people. I don't know if it's this language is Hindi and they're doing rubbish talks here. People are allowed to speak Hindi in my chat. I'm okay with that, but not if it's off topic and if it's spam, if it's anything hateful, if it's anything inappropriate.

[01:38:22]Um, so lock, I'm going to leave that decision to you. You can use the timeout feature. You could time somebody out for 10 minutes, time them out for five minutes, you know, time them out for 24 hours, whatever, whatever feels right.

[01:38:42]Like Hindi is fine in my opinion, but um but not something inappropriate.

[01:38:54]So, like, how do I know my eraser has gotten too small? Okay.

[01:39:07]Okay. So you're saying x - c less than epsilon / 1 + 2 absolute value of c I don't how would you know that like we don't know that c is we don't know where c is C can be a positive number. C can be a million, right? Or C could be really tiny like 0.00001 and Oh, wait. Oh, then but no matter what, we're still dividing and getting a Okay. Okay. Yeah. Yeah. I mean, this would be true.

[01:40:07]Yeah, this this would be true. I see that. But how did you get there? How did you find that also? Like okay, X - C, it's less than delta, which is less than or equal to epsilon.

[01:40:30]Therefore, x minus c is less than epsilon which is less than oh wait just because x because x is near c within epsilon precision.

[01:41:01]I mean like this one is much smaller potentially.

[01:41:08]It should be sandwiched in between these two. Right?

[01:41:18]So my assumption that it's less than delta only tells me that it's less than epsilon, not less than the smaller thing that you just told me.

[01:41:33]My internet doesn't want to let me help RIP.

[01:41:44]Okay, I don't mind if you all talk with each other and ask what you're eating.

[01:41:48]That's fine with me. Just ignore it, I guess. Um, but if there's not a mod who speaks Hindi in the chat, then I'm going to ask that you all keep it unrelated. I mean, yeah, lock I feel this is a decision point for, you know, for us like, do we allow off-topic conversation? I mean, lock is a mod, you know, and it matters, I guess, what what kind of design we want of the of the chat. That's okay with me if it's just if y'all are talking about what you're eating. That's okay with me. But yeah. All right. So, all right. Stream is fine. I'm going to just walk and grab some water real quick and come back. I could ask Gemini, but that's lame, man. That's so lame. This is just one of those places where I'm like, "Oh, this is I'm like kind of stuck. I want to just like pace, go grab a glass of water, come back, and then it might come back to me. Like sometimes you just get kind of stuck and then you're like, I need to reset, like get some fresh eyes, just go for a brief little walk. Write one in place of epsilon, but there's no epsilon right now. Like, um, be right back. Yeah, be right back.

[01:43:25]in the inequality. Well, I don't have like we're trying to prove that this thing is less than epsilon.

[01:43:35]I I don't I don't think that that follows.

[01:43:39]Um Cyber, I think that was Leo lectures.

[01:43:42]Leo lectures recommended that one to you, Cyber or Potato. I should call you potato, not Cyber.

[01:43:50]Um, okay. Thanks everybody. I'm going to be right back. Here is some classical guitar recording that I recorded a little while ago. I hope you enjoy it.

[01:44:00]Be right back.

[01:52:16]Hey, hey, let's get back to it. Uh, sorry about the lag. It looks like my computer ran out of space and uh, yeah, apologies for the ad. We got to start running ads. Um, thanks for watching.

[01:52:31]Let's get back to this. So, I am trying to finish this uh delta epsilon proof. Epsilon delta proof.

[01:52:39]Levi, how do you feel about your epsilon delta proofs? I'm so not great at them, man. Like, I'm just so not great. Um, we we're trying to prove Oops. Oops. Oops.

[01:52:58]We are trying to prove that the limit of X2 is C^2 as X tends to C.

[01:53:41]Um and from here we define an epsilon neighborhood.

[01:54:06]Which then gives us It should be more like this, shouldn't it? Yellow line up here for epsilon.

[01:54:14]Blue line.

[01:54:18]Blue lines down here for delta.

[01:54:40]I mean I really think that we can take delta to be epsilon and it should work.

[01:54:57]But in the in the proof we can also bound it by one. So delta must be either epsilon or one. If epsilon is bigger than one, then well, yeah.

[01:55:25]Nice. Okay. You didn't get an ad. Great.

[01:55:27]I think that we still get I think that we still um Oh, you don't think delta equal epsilon would work?

[01:55:42]Is this one of these where I should just say like magically like take delta to be the minimum of one and like something like epsilon over two like split epsilon by two just magically?

[01:55:59]I mean wouldn't this guarantee it? Like it's just that the shapes like drawing the graph I feel like that distance matches this distance.

[01:56:15]But maybe that's just in this example. Let's let's make this a much more extreme example.

[01:56:32]So if we take a point where it's growing much more rapidly.

[01:56:43]Ah yeah I think you're right. Yeah the delta equal epsson I think won't work.

[01:56:51]You're right. If this is growing much more rapidly over here. I mean, that's an extreme, but still up here.

[01:57:04]Um, say that this is the x value C.

[01:57:27]And then this is the function value C^ squared and let's say that I took epsilon to be one.

[01:57:54]So it's not like a tiny tiny tiny neighborhood.

[01:58:00]It's a relatively actually large neighborhood around then delta must be much smaller than one. You're right.

[01:58:20]Yep.

[01:58:28]I mean, these are not to scale, but yeah, you're right. Delta must be quite a bit smaller than epsilon.

[01:58:42]Hm.

[01:58:51]H.

[01:58:59]Yeah, the handdrawn graph in that first example was a bad spot, but now this one demonstrates much more clearly because I picked a place that was where the slope was similar to um y=x, right?

[01:59:14]The tangent line was similar to y= x, I think.

[01:59:21]No, that's not the point. My graph is the point is that this is the graph of f ofx= x2 cy. So in the epsilon delta definition of the limit um for all x in the delta neighborhood. Good.

[02:00:07]Um then the function value is extremely close to the limit value.

[02:00:26]So the blue line is this the blue lines are this delta neighborhood.

[02:00:32]Um, and then every point within that blue rectangle, like for every point, for every x value in this little blue rectangle, the y values are within our tolerance, our allowed tolerance. So like C^2 C^2 + epsilon and C ^2 minus epsilon that's what I mean by epsilon neighborhood.

[02:01:11]So it's similar to L. In other words, it's in the epsilon neighborhood.

[02:01:29]Yeah, the expression for delta should be obvious and we're all just not finding it somehow.

[02:01:41]Well, I feel like Delta is um I think Delta is Maybe epsilon squared because if delta equals epsilon and epsilon is small then epsilon squared is going to be smaller Right?

[02:02:23]And of course one is equal to 1^2. So maybe this is easier to think of, right?

[02:02:36]1 / square root of epsilon.

[02:02:44]My guess is epsilon squared would be simpler to try.

[02:02:56]Like that's my favorite way to make a small thing smaller like square it.

[02:03:02]Um, and yeah, minimum of epsilon and one is is epsilon. And then that fails.

[02:03:35]Epsilon squar is only smaller than epsilon if epsilon is greater than is less than one. Yeah, but if if epsilon is Oh, wait. Shoot. No, here I did take epsilon greater than one.

[02:03:56]Oh.

[02:04:12]I mean that's where I thought that um Yeah, that's not going to work, is it?

[02:04:25]That's just not going to work either.

[02:04:30]If epsilon is a thousand, if epsilon is like huge and C is sufficiently far to the right, then delta being one doesn't guarantee well Wait.

[02:05:03]Yeah. If C is sufficiently far to the right, then delta being one doesn't guarantee that the Y values are are within the epsilon neighborhood. Right?

[02:05:17]The original question said, yeah, gave the hint to assume delta is less than or equal to one. That's what I'm doing here.

[02:05:26]That is exactly what this minimum does.

[02:05:32]That this allows us to assume that delta is less than or equal to one.

[02:05:52]So, I I found like a a concrete way to make that assumption. Like I can't just assume it without defining it. Like finding a way to um Hey, take care Levi. Good to see you, man. Good to see you.

[02:06:16]Oh, Levi. Oh, if if you don't mind, I'm going to write your name, by the way, in this thanks page.

[02:06:22]Maybe I did that yesterday already.

[02:06:24]Thanks, man. Thank you for helping us get here, you know. Um, so if anybody signs up as a channel member, um, you get your name in the thanks page or even a super chat. Super chats also get your name in the thanks page, any any amount.

[02:06:44]Um, I'm just like I suck at analysis. I've always just It seems like magic. Like once like you need to have some clever idea of what to set your epsilon as.

[02:07:08]Um, so usually you do these by working some scratch work. I should really have that one guy on. There's that struggling grad student. Man, I would love to have struggling grad student on stream to help me with these problems cuz he's like really good at analysis. He's just like a nice guy. Like from what I can tell, he just seems like a super nice guy.

[02:07:33]Um, anybody out there watch him?

[02:07:35]Struggling grad student is really good YouTuber. He's not a streamer, but he posts videos.

[02:07:42]um would love to have him on as a guest.

[02:07:46]I wanted some teacher who would teach me good quality problems like every topic, but I found no one in high school teachers used to spoon feed me. And you're also DVM. All right. Good to see you again. You should, you know, just get to the one account that has the channel membership.

[02:08:04]Uh okay. So often you do scratch work first.

[02:08:17]So the scratch work is that we want x^2 - c^ 2 less than epsilon then then um man I just keep coming to this I keep coming to this feeling Oh wait, if that is less than epsilon we don't know that I can't split it as a um Koshi Schwarz inequality because you used to watch struggling grad student. He was great.

[02:09:29]I just didn't follow your argument.

[02:09:33]I'm sorry. I just did not follow it. Um, got some values. That could be a w to find some pattern.

[02:09:48]Okay, we need X C epsilon X2. squared delta f of um f of c minus delta and f of yeah plus and minus delta. Let's just find both of those values.

[02:10:39]Dude, there's so many different values to be considering, right?

[02:10:47]I guess I guess it doesn't matter what X is, but if uh C like the extreme examples are going to be far from the zero from the origin. Right? So if if C is 100, we want to show that the limit is 100 squared.

[02:11:21]Take epsilon to be I think we still want epsilon to be small. Let's take epsilon to be one so that we can assume delta is less than or equal to 1.

[02:11:39]Then um c^2 or x2 c^2 is 10,000.

[02:11:58]If we take delta, if we took delta to be one, then um then this would be 99 squared and 101 squared.

[02:12:22]99 squared We can we can do that. 99^ squar is 100 - 1^2ar is 1 2 3 4 uh - 200 + 1 is 9,81.

[02:12:45]9,81.

[02:12:50]Right.

[02:12:54]uh 10. Similarly, 101 squared is 100 + 1 squared is 1 2 3 4 + 200 + 1 1,00 or 10,21, right?

[02:13:15]Um so if if X yeah like clearly this ranges outside of that epsilon neighborhood that we want we want the epsilon neighborhood to be one and so it fails.

[02:13:36]So we need delta to be much smaller.

[02:13:39]I think that the this extreme example is working.

[02:13:46]Um here it's fine. We'll just we'll do both of them like this.

[02:14:14]Okay. So, I like this extreme example that we have. The thing we need to tweak is delta.

[02:14:23]Just out of curiosity, what if we took delta to be 1/2? How much does that help?

[02:14:31]um 99 and 12 squared and 100 and a half squared.

[02:14:47]I don't feel like doing our, you know, magic formulas. So, let's just do 99 12^ squared is 9,900 and 20.25.

[02:15:05]Um, so we got almost 100 closer and 100 and a half squared is Yeah. So, we're getting much closer this way.

[02:15:29]I can't choose I can't choose delta in terms of C, can I? Am I allowed to choose delta in terms of C? Why do I feel? Yeah, 1/2 is still too big. I figured I figured it would be. But we see that we we got a lot closer.

[02:15:48]We like split the the gap in half almost right.

[02:15:55]I want No. So my point right now is that I'm expecting to prove to myself delta equal 1 fails. I was just I was fully expecting that was going to fail. I wanted to see how much it fails and then how much do we need to adjust from there. Like from the graph that I drew before, I convinced myself that for sure like I wasn't I wasn't thinking that that was going to work. I wanted to see this fixing, right? We're going to keep this extreme example.

[02:16:35]Um am I allowed to choose delta in terms of C?

[02:16:41]Is your number theory playlist enough for problem solving basics? None of those are designed to learn from. None of those are designed to learn from. So, I wouldn't recommend it. I I want to make that really clear. There's they're very unstructured. Like, I am hanging out practicing trying to teach out loud, but they're not they are not um a course. None of those are courses. So, watch out watch out for that. Um am I allowed to Dude, I don't know if this is cheating, but in a dead I should just look at the book before I ask the stupid AI.

[02:17:23]Like, I have a quick question. Like, am I allowed to choose C in my am I allowed to use C in my definition of delta? Cuz then I think that I could get something that works. Do we have any examples in the book?

[02:18:08]I don't remember ever defining delta in terms of C. So I don't think so.

[02:19:00]So, how can I figure Ow!

[02:19:33]Yeah, that's why I'm thinking like, wait a minute, C is a fixed number throughout the entire proof, right? Like if that's that's my thought, too. like it.

[02:19:57]Uh I'm mixing I'm mixing all sorts of things right now. Oops.

[02:20:03]Right.

[02:20:04]I think we can I think we can cuz it feels like I would need to choose delta in terms of C here. So what happens if I use one over 100 times epsilon because epsilon is Oh shoot Hey.

[02:21:25]All right. First, yeah, I think I can formalize that. But if I choose delta to be 1 over 100, then f of um 100us 100 and f of 100 + 100 is 99.99 squared and 100 + 100 squared which is We're getting closer. We're getting so much closer, but it's still not right.

[02:22:39]99 9,998.0000 0 0 0 1.

[02:22:52]So, we're still outside of the epsilon neighborhood, but we're we're getting really really close.

[02:23:38]Can I always just take a half? Just also take a half here.

[02:23:44]Um, so what I'm suspecting, this is really gnarly, but I'm like I'm thinking delta is 1 over 2 C minimum of 1 and epsilon. I just feel like this problem is not expecting something as crazy as this.

[02:24:12]This seems like a little Yeah, exactly. See, right. Hey, thanks, dude. Charizard Pandy, there's somebody else in the in the Discord who's a Charmander, so you all have to meet.

[02:24:27]Um, if you just watch to get an idea and then read the book yourself, will I be able to understand? That's the goal. But the main series that everybody's subscribing for and that I'm getting a lot of attention for the main series is I am making lecture videos out of this book. So for what value of x does f(x)= c^2us epsilon between that X and C.

[02:25:28]Y'all see why? Like I always thought that analysis would just kick my ass in grad school. I just I don't know why I've never been able to really do great at analysis proofs. Like these things are just magical to me. Um Hik.

[02:27:00]I I think that this one works.

[02:27:06]Uh, plus one over Let's see. Wait.

[02:27:17]Yeah, this one works.

[02:27:53]So it is within in one unit.

[02:27:58]I chose epsilon to equal one.

[02:28:02]Right? So now finally we've got we have found one neighborhood.

[02:28:12]Uh here wait I know how to do this. Cut this one. Move this one.

[02:28:21]I want it to match the others. Minus is above, plus is below.

[02:28:29]So now we found one that works.

[02:28:35]I'm using my handy dandy calculator.

[02:28:38]Um, one over 200 did not work.

[02:28:44]One over 300 did work. One over 250 does work, I think.

[02:28:52]Yeah, one over 250 does work. One over 275, we're doing a binary search.

[02:28:59]One over 275 works.

[02:29:06]One over 285 works below 1 over 290 works below.

[02:29:16]One over 295 works below one over 299.

[02:29:21]works below.

[02:29:24]Wait.

[02:29:26]Oh, wait, wait, wait. Oh, I'm going the wrong direction. Wait. 250 works. 200 does Oh, wait. One over 200 does work. Shoot.

[02:29:48]Oh, no. No. one over 200 works. Okay, cool. Cool. Here's some neat little data.

[02:29:58]This is a really interesting point.

[02:30:02]This is valuable. Even if it's not going to yield a proof, I still think that this is valuable to build up some intuition on this thing.

[02:30:18]Shoot.

[02:30:21]Okay.

[02:30:28]1 over 200 is really interesting because f of 100 plus 1 over 200.

[02:30:50]In other words, that squared One of them works, one of them doesn't.

[02:31:02]And I've already forgotten. I think Yeah, minus works.

[02:31:25]It's above that.

[02:31:33]Yeah, it's how do I present it? One, two, three, 4, 1, 2, 3, 4, 25, 5. So, this one is within one unit of 10,000.

[02:31:52]So, it works.

[02:31:57]Um but then the other one does not 1 2 3 4 1 2 3 425 1 2 3 4 25 This one does not work.

[02:32:18]So delta needs to be smaller than 1 over 200 which is now depending on the value of C I believe.

[02:32:36]And now I'm just going to check uh one over 250 works above and we assume like since the previous one worked below then all of these are going to work below.

[02:33:03]Uh, so one over 200 failed above. One over 250 works.

[02:33:12]Oh, wait. No, one over 200 failed above.

[02:33:15]One over 250 works above.

[02:33:19]So, what about one over 270? Oh, wait.

[02:33:22]No, 225.

[02:33:27]I'm trying a binary search here.

[02:33:34]one over two 25 and we're checking above.

[02:33:44]Yeah, this one works. I just want to see if I can find the point like within a with a little more accuracy.

[02:33:56]When does this flip?

[02:34:01]I don't think I need any of this anymore.

[02:34:09]In other words, I'm saying that my test that I'm testing is f of uh c plus delta is less than L + epsilon specifically in in this example that we're working 100 plus delta squared less than 1 10,01.

[02:35:09]This is the test that I'm running right now and trying to find a point when it fails. So halfway approximately between 200 and 225 is 212.

[02:35:27]I'm just at this point I'm just super curious. I don't know if there's a way that this is going to yield an answer.

[02:35:34]This one also works.

[02:35:39]This one also works.

[02:35:52]This one also works.

[02:35:59]And this one also works.

[02:36:13]This one also works.

[02:36:20]I think we've got 1 over 2 C.

[02:36:30]Yeah, this one also works. It looks Yeah, it looks like we're going to be going for Yeah. everything. Let's see. I can just throw some zeros at this thing. Oh, no.

[02:36:41]If I throw zeros at it, we do get it to fail eventually.

[02:37:05]which is interesting.

[02:37:08]I got this one to fail.

[02:37:12]0.001 O 01.

[02:37:25]This one fails.

[02:37:29]Unless it is a some kind of like rounding error.

[02:37:37]Unless it's some kind of rounding error.

[02:37:39]I'm getting 10,01 plus a small. I mean, I don't I don't think that this would be something that the computer would fail at.

[02:37:56]That's interesting.

[02:38:00]If I take away one zero, it it works.

[02:38:13]We went into binary search mode here. We went into gather some data and find a point when true flips to false. And we just found we finally found a place where it fails.

[02:38:37]So what about right in between the two? In between the two works.

[02:39:00]What if I split the difference between those? That one works.

[02:39:12]Um, did I get this wrong?

[02:39:29]Wait. O1.

[02:39:32]Did I Did I get something wrong here? O Yeah. O1 fails.

[02:39:39]O1 fails. O2 fails.

[02:39:45]O3 fails. O4 fails. O5 passes.

[02:39:54]And then anything above 05 passes.

[02:40:09]So, I I went the wrong direction on this one.

[02:40:16]I saw one over 200.4 fails.

[02:40:28]One over 200.45.

[02:40:31]045 fails.

[02:40:46]One over 200 49 fails.

[02:41:11]If I keep adding nines, it takes a bunch of nines until we finally get it to pass. And now we're talking about freaking rounding errors, I feel.

[02:41:30]This has got to be some kind of rounding errors, floating point errors here. But 2004999 fails.

[02:41:46]Whereas 1 over 200.499 So for 9 9 uh nine if I add another nine to this one now it passes.

[02:42:08]Uh so how much do I trust this?

[02:42:11]I feel that this is like I'm just kind of analyzing floating point rounding errors.

[02:42:27]You got to teach me. We can trust O5.

[02:42:36]Hey Leo lectures, what's up dude? Leo, what up, dude?

[02:42:46]Yeah, please do teach me some math, Leo.

[02:42:48]We got to do that again. One over 200 does work. Did I I Did I catch that eventually?

[02:42:58]I saw one over 200 fails above, but it passes below. Did you Did I get that wrong? Yo, Divium, thank you again.

[02:43:09]Epsilon is one. I know that um above no look 100 + 1200 quantity squared is Oh, I I wrote that wrong. I wrote this down wrong. My bad.

[02:43:39]This was 10,0001.0025.

[02:43:45]My bad. That I wrote it down wrong. But this one This one failed.

[02:43:51]Yeah. Yeah. That's That's my fault.

[02:43:59]So, just in case people are curious, what I did what I did was I found that according to according to Steve Jobs, according to my Mac, this is 10,000 and one point one two three four like here dot 0 dot dot dot a bunch of zeros and then 44 right so that is more than one away from 10,000 whereas Whereas if I add one more nine to this thing now this is 10,000.999 Nine.

[02:45:27]How many nines are there? 1 2 3 4 5 6 7 8 9 10. There are 10 nines in this one. And then in the previous one there were one, two, three, four, five, six, seven, eight. There were eight zeros.

[02:45:54]Okay, so this is the test and we finally found a point and that's good enough for me right now. Um, so is this telling me that this needs to like either I'm just analyzing floatingoint errors and something like 1 over 2 C epsilon or some or some freaking like that? Um, like wouldn't it be like 1 over 2 C * like epsilon over 2 C or something or Yo, Chi the actual delta is some irrational maybe. I'm just I was just curious, dude. I I just wanted to see according like I got a binary search or I got a little test and I was just curious at this point to take a break really and see how close to 200 do I need to be to 200.05.

[02:47:22]So it does sound like maybe it is some irrational and it's not just something as simple as this.

[02:47:30]So how can I how else could because in in this example C is 100 and 1 over 200 is extremely simil and epsilon was one.

[02:47:48]So, um, bro Chewy, thank you so much for joining as a channel member, man. It really means a lot. Um, Choy 820, dude, thank you so much for joining as a Luna member. I'm going to write down your name on this problem that I'm currently working.

[02:48:26]Where is my pen?

[02:48:31]Uh, Choy 820.

[02:48:44]Good. Seriously, thank you so much as for joining as a channel member. You're right there, dude. Appreciate you.

[02:49:02]Divium. Um, I am starting like my number three playlist was on Andrews, but I didn't get that far. So, I I really don't want to promise too much for you.

[02:49:15]about how how much that playlist is going to help you. Like don't don't try to treat those playlists as a way to work through a book. Just like yeah, don't don't try to treat those playlists as a way to work through a book. Like how effective would this stream have been for learning epsilon delta?

[02:49:38]It's not it's not a very good way to learn epsilon delta. Um, okay. So, yeah, we did get stuck on a while loop and I wanted to find it. Um, we basically found it. So, I think I can choose Can I do something like this?

[02:50:36]Like none of the examples in the book define delta in terms of C, but it seems reasonable to me.

[02:50:51]Like we may just be hitting some kind of rounding errors cuz it's so close to 200.

[02:51:13]I want to see if chatpt can give me a hint on this one.

[02:51:19]In an epsilon delta proof, can I define delta in terms of uh epsilon and c.

[02:51:36]Yes, that's what I thought.

[02:51:39]That's what I thought.

[02:51:43]C is fixed before you choose delta. So it may be a function of both.

[02:51:48]Yep. And it's using exactly they're doing exactly this proof too.

[02:51:55]The the the example that ChachiPT is giving me is this one. So let's see.

[02:52:03]We get that restricting delta less than or equal 1 gives X minus C absolute value less than one. So we get okay wait. Oh, so we do.

[02:52:16]Okay. X + C.

[02:52:21]Yep. Yeah. Yep. Yep. So, this is apparently like a standard example or something of this. Okay.

[02:52:29]So, many times through we've been working from the scratch work like this.

[02:52:41]Um this is less than or equal to x + c x - c and this is less than or equal to x + c.

[02:53:17]Assuming delta is less than or equal to 1.

[02:53:33]All right. So we got here before and then I had this idea before that this would be the same as x - c + 2 c. I had that idea and then this is less than or equal to x - c + 2 absolute value of c.

[02:53:57]Right? So we got to this point before.

[02:54:02]This is a triangle inequality.

[02:54:56]So if we defined delta as this.

[02:55:53]Now we finally have a chain that ends up being less than or equal to epsilon.

[02:56:00]So the step that I'm doing right here is that delta I just chose was epsilon minus 3 absolute value of c. But I don't know that that is I don't know if that doesn't seem like a valid move because that's not guaranteed that's not guaranteed to be greater than zero, right? So I don't I don't think I'm allowed to do that one.

[02:56:38]Cool.

[02:57:16]I don't think that that move works.

[02:57:20]I'll answer questions like once I'm once I'm done with this.

[02:57:26]Um, I mean by um delta is we could just choose delta to be epsilon 1 - 2 absolute value C, right? Yeah. Plus two absolute value C.

[02:58:32]Yeah. What am I doing then? Now we just have epsilon.

[02:58:37]Okay. And now do we have a strict inequality anywhere along the way?

[02:58:43]Um xus c should be strictly less than delta.

[02:58:50]It should be less than or equal to delta. Um x Yeah. absolute value x minus c is strictly less than delta. And then finally therefore um f ofx minus l is less than epsilon.

[02:59:08]Cool. Yeah, there it is. Okay, so now we could restart the whole proof uh and say let x um let epsilon be greater than zero um and choose delta equal to the minimum of one and uh of 1 and epsilon minus 2 absolute value of c.

[02:59:50]Right then.

[03:00:05]Oh, wait.

[03:00:16]No. Uh, shoot. Yeah, we still don't know that delta is greater than zero.

[03:00:30]Yeah, we still have the same problem that I was concerned about before. Um Yeah, like when C is 100, then this just doesn't work. It doesn't make sense or it fails.

[03:01:02]H I'm just going to read Chachi PT's answer. We We got pretty damn close on this. We're so close.

[03:01:09]Yeah, it's nonsense, right? Yeah. Yeah.

[03:01:12]It doesn't work. That doesn't work. It's a negative value. It's not greater than zero. Doesn't work. That doesn't work.

[03:01:20]Um, I saw that, too. Yeah, that doesn't work.

[03:01:31]All right. How does this go?

[03:01:44]We get x + c.

[03:02:06]All right. Chad chippity gives me x^2 - c^2 is equal to absolute value of x - c * absolute value of x + c restricting delta less than one less than or equal to one gives x absolute value of x minus c less than one. So absolute value of x plus c Oh yes. Okay. Absolute value of x plus c. They are doing the same trick that I was thinking. x + c - 2 c. Oh wait no x - c + 2 c.

[03:03:00]And then this by triangle inequality strictly less than 1 + 2 absolute value C.

[03:03:20]Right.

[03:03:27]Yeah. So, we were super damn close. Um, they're saying then, um, choosing delta being the minimum of 1 and epsilon over 1 + 2 absolute value C.

[03:03:52]Oh boy, works. Here delta explicitly depends on both epsilon and c and this is completely standard.

[03:04:05]The only thing delta cannot depend on is a particular value of x since x is quantified after delta is chosen.

[03:04:16]This man I don't know um this is this feels like a hard problem for a thinking forward.

[03:04:24]Look at your graphical I didn't follow your graphical explanation. I mean I was thinking that same graphical explanation. I just don't see I wouldn't like I wasn't seeing like a way to get an algebraic you know like the graphical explanation was from what I read was like quite similar to what I was drawing here and yeah I I really had a feeling that it wasn't going to be a square root and that I wasn't going to find an algebraic solution. here based on a based on a graphical explanation any further. I had a feeling it wasn't going to be a square root thing and it wasn't. So, yeah.

[03:05:20]So, we're very close. This is extremely close to what I was that that I found with this um process. So, I should have just noticed that all of these were to see, you know, all of those.

[03:05:43]Yeah, I could have thought to just add one and then all of these would have like this would process would always work. That's kind of cool. I mean like we we at least got to delta is extremely close to 1 / 2 C. And it turns out that we can choose delta is 1 / 2 C + 1 and it would satisfy this. So I was I was damn close. And then somewhere along the way I thought yeah I believe that we can see look at this. I was so close. epsilon over 2 C.

[03:06:29]This is what I thought. This was like my hypothesis.

[03:06:33]We just were missing the fact that if I just added one to all of them, then that would satisfy all of these cuz like some of them failed.

[03:06:47]Like I was I was trying to figure out what is that like more precise bit and it would be it would be something that depends on C and it would be complicated I think.

[03:07:08]Oh, your thing finds the tightest possible bound. Okay. So then your thing you think that you can figure out what this is, right? You're saying that because I I think right now in my head that this is 2 C plus some function of C I believe.

[03:07:36]Um Yeah, it doesn't it I' I've already proven that it's not the tightest bound, right? Like through this evidence that I thought was valuable and it is valuable, I can see that it's not the tightest bound because some of these are tighter than 1 over 2 C. It's 1 over 2C plus something. Um, and I think that like I guess I don't I don't care, right? What?

[03:08:34]And then this was this was the other thing that I was missing. I was missing that it should be absolute value of C and it's where I I paused. I was like, "Yeah, but like what if C is negative and other things." So now I have some intuition a bit on sometime like sometimes I might be able to just add one and and we would find it.

[03:09:03]Divium why did I choose this book? This chooses this book is excellent. It's both conceptual and um applications based. So on the one end uh we have Stewart very popular on the other hand we have spac uh Stewart is like applications or computations calculations no proofs speedback is all proofs essentially I really think Tolman is great because it gets both proofs and calculations.

[03:09:50]It's a great book. It's really nice. Um, let me read your questions, Diviian. Um, where can you find math like number theory a book? Dude, I I feel like I answered this earlier. Um, I don't know like a if you're asking about videos or something like eventually we will be adding more and more number theory in this series, but it's going to take a while and you're going to learn some number theory in my abstract algebra um in the abstract algebra streams which are going to begin in a couple weeks.

[03:10:28]Um, I don't know. Uh, I really like so far I really like Andrews as a nice number theory book. Um, I like Andrews and I like for number theory like I can't cover every topic every stream. You know, there's too many subjects and I'm never going to get anything done if I just if I work all the topics. I I need to stay focused and work one subject at a time or three subjects at a time. Um Andrews is a nice cheap do. Um I used rosin um for my undergrad book and and it's it's pretty good.

[03:11:15]It's pretty good. It's very friendly.

[03:11:21]Yeah.

[03:11:22]Yeah. That's the name of it. Um, I like rosin for very friendly.

[03:11:38]Andrews is friendly and brief. Rosin is is just it's friendly with a lot of calculations, a lot of problems. Um, if you are uh if you want something harder um you could look at Ireland and Rossin, which is a different roin, which is confusing, but classical intro or wait, modern intro to classical number theory, something like that. That one is That one is hard though.

[03:12:20]Harder. That should be your second number theory book. I don't know. Um Oh, definitely books like Vellamin like any um any kind of book that's like a intro to proofs book like how to prove it for number theory. You could do that or you can always work problems from um like every abstract algebra book is going to have a good amount of number theory in it. Uh like Gallian I think is going to be the undergrad book that we work or dumb it in foot.

[03:13:10]um abstract algebra books will work number theory and we'll we'll do some number theory but it's going to take a while. It's just you know we got to work systematically slowly.

[03:13:25]Um cool.

[03:13:29]I don't know Weissman there's a lot of books out there. Um I'll do number theory eventually. I I will eventually do number theory, but it's, you know, we're not going to focus on number theory for a while. I love number theory. I would like to focus on it, but I think that I mean, I don't know. Like, Lock, do you have an opinion on this? Like group theory or number theory to replace calculus for college math?

[03:14:02]I want to get more intro to proofs and I think it's better to get some group theory out there. That's what my course, my intro to proofs course was taught out of. It was intro to proofs in context of group theory. Um, multivariable calculus. Nope, not for a while. Today was our farewell calculus final lecture. I think farewell calculus. Not not in a while. Um because we're doing we are doing um high school.

[03:14:50]Secondary school math is basic math by Lang.

[03:15:00]We are doing college math and I think it's going to be abstract algebra and I think it's going to be gallion. It might be dumb it in foot.

[03:15:28]And then we're going to do grad graduate school math with representation theory by Steinberg.

[03:15:43]If I add more books and more subjects than this, we're not going to be very effective in terms of developing pure math proof writing skills and having something cohesive and structured. It's really important to not jump around all over the place.

[03:16:03]Yeah, I think number theory is just not that necessary.

[03:16:07]Uh Vob edits, you don't get a blue name because you're not a mod. if we only have a couple of mods. There's just a few mods.

[03:16:16]Um, and yeah, number theory is beautiful, but you can learn it. You can learn what you need in context of abstract algebra and others. So, yeah, vote against multivariable calc because it's uh influenced by vector algebra. Yeah, that's a good point.

[03:16:40]Okay, y'all. I had a I had a blast today. Um, super special thanks to everybody who's here, everybody who's, you know, supporting the channel, everybody who's just here chatting.

[03:16:53]Like, I appreciate you all very much.

[03:16:56]Arcane, what's up, man? It's been a while. Um, I am definitely going to hold off on on those others. Um, I read your message before you retracted it. Um, someday we'll do other more analysis topics, but uh, the thing is is that that's not fun for me. Like if I'm studying math for me as Oh, see you in the guitar stream at five. I do I should do a guitar stream, man. I should do a guitar stream. Vob edits, we are laughing. Who's Oh, are we am I in the guitar stream at the moment? What the hell? Am I also multireaming to the guitar stream?

[03:17:41]Wait, wait, wait, wait, wait, wait. Am I in the guitar stream right now?

[03:17:47]Dude, are you serious?

[03:17:50]How did that happen?

[03:17:54]The hell?

[03:17:56]Am I Am I guitar streaming?

[03:18:01]I'm not guitar streaming, am I?

[03:18:04]Okay.

[03:18:06]confused.

[03:18:18]Or yeah, you're laughing as in you're like making fun of me.

[03:18:23]Okay. Okay.

[03:18:26]Uh, hey V, you enjoyed episode 3? I'm happy to hear that, dude. I'm happy to hear that. Okay, I might I might guitar shoot. I need to get better about a schedule and a flow to not neglect the the guitar community.

[03:18:44]I really don't mean to be neglecting y'all. I'm just having so much fun studying math and uh and it's working.

[03:18:52]Came for the cat. Let me get the cat for everybody. Let me get the cat.

[03:19:30]Hello.

[03:19:32]Okay. So, I don't know. She hasn't been a social today. It's surprising. Dude, do you No worries. We should all be practicing guitar instead of watching.

[03:19:40]That's true. And y'all should be practicing math instead of watching, too. Luna. Luna. Okay. She doesn't want to be here today. Sorry. Um uh yeah, arcane. Um for myself, I want to learn algebraic math primarily. I would like to learn algebra to the degree that I can start doing like algebraic topology at some point for myself, but it's going to be a while, I think. Um right now I I I just want to chill in representation theory for a while because I find it just like a really beautiful field. Um, and then like a subject after representation theory that I've always wanted to hit would be Gawwa theory. Like I feel once I feel ready to move on and I want like a new subject in the rotation, it would be rep theory, Galawa theory, rep theory, Galwa theory for a while. Um, and yeah, like I just like algebra. Like after those I'd be like I'd be more interested in in like learning type theory or theoretical computer science over topology like at least for right now.

[03:20:57]But I don't or like algebraic geometry.

[03:21:00]I I would like to learn a bit about algebraic geometry but there's just so much like I would love to learn it all cuz I'm a freaking nerd.

[03:21:10]So yeah. Yeah.

[03:21:13]I just don't know Skittles what you meant by we are all laughing. I'm very like self-conscious about that right now. But yeah. Anyway, thank you everybody. Um yeah, I had a blast today as always. Uh please do like and subscribe and share it with a friend. Um y'all if anybody wants to share this the playlist to the to the math basic math videos. If anybody ever wants to share the playlist or get to the playlist, math can be a hobby.com.

[03:21:53]I've gotten that URL. I've gotten a few other URLs. Turns out I'm the first person to think, let me snag all the URLs with math hobby hobby math. So, hell yeah, dude. All it does is just redirect to the YouTube playlist. So, very sharable URL.

[03:22:19]Can can just say out loud in convo.

[03:22:25]So, if you haven't seen my basic math videos yet, go to go to freaking math can be ahobby.com and we should get some bumper stickers or something. Hi, Luna. Hi, baby. Hi, baby. Epic cattail content. Yeah, I just She hasn't been as social as as like all over the place as I expected, but she will. Oh.

[03:22:55]Okay, thank you all. Thank you all. It's It's been a pleasure. Um, I'm going to take a break and maybe stream some guitar this afternoon. This The stream ran pretty long, so take care. Bye, everyone. Bye bye. And check out episode 3 if you haven't watched it already. It just went up last night and it's all about practice of addition. I have like a few interesting little I think that they're um useful and interesting examples of like addition drills that you can build for yourself that are interesting cool math problems, but they're also just a way to practice addition if you are rusty on addition, but they're pure math problems. You're not adding roller coasters and marbles and baseball hats together. So, yeah.

[03:23:49]Thanks for watching. Bye bye.