Mastering the Art of Reading Proofs: By Example

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We talk about how to read a proof of a theorem in Rudin's Principles of Mathematical Analysis (i.e. Baby Rudin). We show that positive n-th roots of positive reals exists and are unique.
//Books
Walter Rudin - Principles of Mathematical Analysis 3rd Edition - amzn.to/3MDHUis
Maxwell Rosenlicht - Introduction to Analysis - amzn.to/3NFYiAC
Tom Apostol - Mathematical Analysis 2nd Edition - amzn.to/3mwxFlC
//Exercises
- Can you show that if t is a positive real number less than 1, then its sequence of powers forms a decreasing sequence?
//Watch Next
The Real Analysis Survival Guide ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-v5rD0B-zfXw.html
The Analyticity of the Laplace transform ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-FIMkbFQL6XM.html
Introduction to Control Theory ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-0v4WFmOm764.html
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Опубликовано:

23 янв 2023

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Комментарии : 63

@edal7066 Год назад

tell your students to never forget definitions. those are the most important stuff to always keep in mind when building up mathematical stuff.

@JoelRosenfeld Год назад

Very true! So many statements can be quickly dispatched with an appeal to a definition. I'll pass it along!

@sanjursan Год назад

Amen brother. Definitions are the "times tables" of math, the alphabet even. Just commit them to rote. Painful but enormously helpful.

@douglasstrother6584 11 месяцев назад

From Elementary School to Calculus, mathematics is generally all computational. "Compute ..., Solve for ..., Construct ..., Integrate ... ,etc. Higher mathematics are, "Show that all are isomorphic to ." That boils down to contructing sets of and and using the rules of logic to prove the theorem. That's a completely different approach to getting "42" as an answer.

@hellNo116 Год назад

this is why I love and I am mortified of algorithms as a computer scientists. you can write 5 pages and find out that at least 2 of them were pointless if you understood the problem better. this is a great lesson not only for those wanting to learn to prove stuff, but for those of us that want to teach such stuff to students at some point.

@JoelRosenfeld Год назад

I really loved learning how to program for this reason. You start with a massive amount of code, only to realize you can cut it down to a couple of lines. I participated in a programming competition in undergrad. And when I was practicing with my team, I would sometimes reduce a recursion or something else down to a single line. (One time this leveraged generating functions.) It was hella fun.

@hellNo116 Год назад

@@JoelRosenfeld I don't see single line solutions often. I write in c/c++ mostly so most solution are wordy to say the least. however even there the pattern that small code is good code emerges. I mean obviously there are limits to such statement but I think you understand my statement. what I found that is the most important is to not be afraid to create functions to make the code look abstruct is a great strategy, especially if you need to explain the code and or maintain the code base. let alone reusability. the reason I reference the last one is that if you write code the way you made this proof by making many lemmas to prove it makes the code easier to work with and that makes total sense, since you have to prove such code works and that is how you do it.

@coreyevans5734 Год назад

I can recall when I went through intro to real analysis 1, I quickly fell behind. Stress and frustration followed especially with the imposter syndrome I was already dealing with. Funny thing is something eventually broke. I realized I deserve to be at this level because I kept practicing and working over and over and over. I went from failing to passing and now I almost have my bachelor's. Yeah this course is hard. It might take multiple attempts, but you can find a platform to build off of.

@JoelRosenfeld Год назад

Yeah, this is really common for real analysis. I'm doing my best to provide these videos for my students to help them stay oriented and focused. We actually had a day in class where I gave them a hard problem to work on together. It was intended as a wake up call well in advance of the exam, so that they can start reforming their study habits.

@jikan6975 Год назад

Main takeaway from the thumbnail is that a stroopwafel and a cup of coffee is required, will keep in mind.

@JoelRosenfeld Год назад

I may or may not be eating a stroopwafel right now... >.>

@mmariokart231 Год назад

I’ve always loved analysis, and love this channel because it explains the more vague and disheartening experiences and spaces we find ourselves while pursuing the subject, makes me feel like I’m on the right path while giving me ways to practically navigate through it, can’t say thank you enough for taking the time and care to put this out there for those of us still fumbling around in the dark!

@JoelRosenfeld Год назад

I’m happy you have found it helpful! Even after all these years, I often feel like I am poking around in the dark. Lol it’s so satisfying when something does come together

@zoedesvl4131 Год назад

I remember reading an article by a PhD explaining some concept and did a step-by-step verification that took several pages. But the verification on the textbook was "by properties of ..., it is ..." in a single line. Yeah by properties of elliptic curves, Fermat's Last Theorem is true so there is no need to write a paper of 100+ pages lol.

@JoelRosenfeld Год назад

Lol, I guess some things are best left in the black box!

@Bunnokazooie 3 месяца назад

My favorite of your videos!

@CalBruin Год назад

I like your demonstration of rigor in doing a proof.

@meteor8076 Год назад

Hello, do you know - what would be the easiest path to get into stochastic calculus, knowing only elementary calculus (at the level of Stewart's book) ?

@Cyclonus-fc1xx Год назад

Really good video!

@JoelRosenfeld Год назад

Thank you!

@mircopaul5259 11 месяцев назад

I feel like some sort of visualization and just playing around with examples is what allows me to build up intuition for an object/definition/theorem etc. Sometimes some direct proofs and constructive proofs inherently convey "the intuition" behind a result. Indirect proofs/proofs by contradiction usually don't help much with intuition, showing that the complement of a statement can not be true often doesn't really give away too much about why the original statement (the complement of the complement) is true.

@douglasstrother6584 11 месяцев назад

Agreed. Showing each algebraic statement on a number line would help in developing a mathematical "knack". The overlapping (or not) of line segments would give a visual representation of the theorem.

@prathamesheedigi Год назад

Can u make a video on Basel proof. A video on weirstrawss factorisation theorem

@jennifertate4397 3 месяца назад

Thanks. I plan to watch this a few more times at a lower speed. I have Baby Rudin.

@JoelRosenfeld 3 месяца назад

Sounds good! Let me know if you have any questions.

@jennifertate4397 3 месяца назад

@@JoelRosenfeldThanks for the offer, highly appreciated.

@ajaymehra502 Год назад

I have completed my bachelor's degree in physics, Now I want to switch completely to math research, I am struggling with the proofs thing in math, But I am trying my best. In May, I will be writing a exam on maths for my masters degree.

@JoelRosenfeld Год назад

It's a difficult transition, for sure. Just stick at it, and read and practice. Lots and lots of practice. Good luck!

@CalBruin Год назад

I hate baby Rudin. One reason is the proofs are too smooth, in order for the text to be terse. Terse writing leaves out a lot of missing steps.

@jennifertate4397 3 месяца назад

I agree with the last 2 lines.

@CalBruin Год назад

When cutting my teeth with baby Rudin the first time, our instructor said proofs is matter of unpacking definitions. I understood that intellectually but not internally until years later when reading a rather good book on how to do proofs, when it advised how to think in tackling a propostion. In the chat exchange clip, there were the suggestions relaxing rhe hypothesis, making the hypothesis more strict, play around -- that there is key help. Play, yes play with the given problem to see how it works. A takeaway, going back to definitions, is translate the given proposition into plain language. What is the theorem saying by translating every part into the base definitions.

@JoelRosenfeld Год назад

That is very good advice. So often, a theorem really just comes down to looking hard at a definition.

@CalBruin Год назад

Intuition arises from experience. In other words, whe. one does something enough, they develop a habit that guides them through in tackling even something unfamiliar.

@JoelRosenfeld Год назад

Absolutely true. Intuition takes a lot of time and practice.

@joechakib3948 Год назад

For the first implication, when proving the statement (y1 < y2 ---> y1^n < y2^n) could you have divided y1^n by y2^n which would have given (y1/y2)^n < 1?

@JoelRosenfeld Год назад

Yeah, that could work. You’d still need to go through the induction process, where you multiply one side by y1/y2 and the other by 1. It is ultimately the same argument. You just exchanged y1 for y1/y2 and y2 for 1.

@joechakib3948 Год назад

Ahh okay I see. Thanks. This was actually my first RU-vid comment. Best of luck to you. Love you content.

@JoelRosenfeld Год назад

@@joechakib3948 Happy to have you here! Please feel free to comment whenever you have a question.

@adnanhashem98 11 месяцев назад

If you expect the reader to come up with an idea for a proof, as an author, you should share your thoughts and scratch work that lead to some of the proofs in the textbook. To better understand what I mean, pick up 2 books on Real Analysis to see the difference. The first is Baby Rudin and the second is Real Analysis: A Long Form Mathematics Textbook.

@JoelRosenfeld 11 месяцев назад

When you start doing your own work, or are needed to evaluate someone else’s, it’s a really important skill to be able to find the gaps and where more work is needed. That’s exactly what Rudin trains students to do; to find the unspoken pieces of the proof that still need to be verified. If a student still has trouble, then their professor can help fill in the gaps, and heck, that’s what this video is here for too.

@adnanhashem98 11 месяцев назад

@@JoelRosenfeld I agree that it's an important skill to be able to detect incomplete arguments or even incorrect ones. However, I suppose (from you're reply) that you are assuming students who use this book (or similar books) have access to teachers who have enough experience to give them the nudge when needed or to correct their erroneous/incomplete proofs. However, that's not always the case. In fact, I am interested in such math subjects, and I try to learn them mainly using textbooks. I assume Real Analysis is one of the first courses a math student typically learns to improve their skills in writing proofs. So, I find it more logical to include the ideas of how one would come up with the proofs and motivate the definitions and say why it was introduced (which is the way the second book is written) rather than polishing the proofs just occurs to one mind without trying to make some illustrations etc. I'm not in the math level to criticize Rudin's book, and it's just a humble opinion from the perspective of a self-learner (who do not take real analysis as part of a university course).

@JoelRosenfeld 11 месяцев назад

@@adnanhashem98 but you do have access to teachers to help guide you. You are talking to me right now. The internet is full of places where you can ask for advice. Stackexchange, r/learnmath, etc. professors and students both hang out at places like that.

@adnanhashem98 11 месяцев назад

@@JoelRosenfeld Maybe I'm not yet used to asking questions on such platforms. I guess I need to give it a try and see how things workout then. Thanks for the advice.

@victorsauvage1890 5 месяцев назад

Am unable to see the terms you are pointing to. How could this explanation be improved? Adopt the same gentle method which you would apply in teaching someone how to drive a motor vehicle.

@RSLT Год назад

Cool

@AxiomTutor Год назад

Long story short, can't teach it, you just pick it up. I agree in a philosophical regression-argument sort of sense. You could take any step in any mathematical proof and have a student ask: How did you know to do that? You can furnish some answer perhaps, but it can always be met with "Well but how did you know to think of THAT?" And if you can keep furnishing answers, the student can keep responding to each with a new "How did you know to think of that?" At some point an explanation has to ground out in "It seemed to make sense." Still, I think a lot more explanation is possible than is typically given by mathematicians, either in class or in textbooks. I somewhat suspect that if we taught mathematics more historically, showing the progression of how we discovered these ideas, rather than trying to package the ideas into concise reference texts that we race undergrads through, people wouldn't be so confused. In your example, one could ask "How did you/Rudin know to form the set of all t: t^n < x?" Well, you just learn it over time. "But ... that can't be the whole answer. Someone discovered it first, and then other people got used to the trick. How did the first person to do it, know to do it?" And now you have to dig deep into the historical context, to understand what did they know and when did they know it.

@JoelRosenfeld Год назад

I do try to talk about the historical development of a bunch of ideas in my class. For example, in my last video about the square root of 2, I mention how clever Rudin was to select that function he used. Now Rudin's function IS pretty clever, and more clever than what was done first. But if you look at Dedekind's work, he used a more complicated function that essentially did the same thing. Rudin just benefitted from a century of refinements. I don't know for certain about the n-th root business. But if I had to peg someone for it, it would be Dedekind again. His definition of the reals really emphasized the least upper bound property, and it would naturally lead to definitions like the one we see here.

@veil6666 Год назад

"I recently had a several hour long argument with a student" bruh

@JoelRosenfeld Год назад

It was several hours over Reddit. Basically an extended chat session. I don’t know if I got through to him, but I was hoping the time I was putting in would help him.

@rafaelsuarez3059 Год назад

@ThatMathThing I really like your content but the music is too loud and interferes with intelligibility.

@JoelRosenfeld Год назад

Got it. I'll work on improving on that. Take the volume down a notch.

@kevinthompson9953 Год назад

@@JoelRosenfeld Or consider eliminating the music. I find it highly distracting to listen to music while you are also talking. Not trying to be critical, love your channel and your efforts.

@jackbrolin7709 Год назад

@@JoelRosenfeld I second what's already been said: consider removing the music altogether. A few seconds at the beginning can work well for an intro, but can become very annoying very quickly

@mohad12211 Год назад

@@JoelRosenfeld This type of music is designed to keep your mind occupied due to its various notes, which is exactly the opposite of what you are doing, a proof related to analysis. Consider using more calm music, or removing the music at all, maybe try one then the other and see which one is better.

@JoelRosenfeld Год назад

Ok guys, I'm giving this a try. In the next video I'm cutting the music on all the board work, and reducing the volume overall. I definitely hear you on the music. I have other people telling me that they like it, so it's hard for me to really pick between the two. One thing that is really interesting is that RU-vid is experimenting with having multiple audio tracks for their videos. The intention seems to be to consolidate multiple languages into a single video. But if I could have a "with music" and "without music" track, then that might be able to make both parties happy.

@solaris413 Год назад

please don't add such music in background it hurts my ears

@JoelRosenfeld Год назад

It gets better with the subsequent videos. Took me a while to figure out, but I feel like I’m doing better with it now

@douglasstrother6584 11 месяцев назад

@@JoelRosenfeld No music is probably better, but I will miss your earlier vids with "Creedence," "The Rolling Stones" & "The Who" blasting in the background!

@TheJara123 8 месяцев назад

Though I appreciate your efforts I think your presentation style is really bad for these topics in math...higher math, abstract levels require proper presentation....and slow pace....but you present them as though we know it, but we dont!! so take one problem and work it out from beginning to end, along the way share your insights and passions....

@JoelRosenfeld 8 месяцев назад

I appreciate the input. Longer videos are harder to sell, and take much more time to edit. These videos generally take up to two weeks for 10 to 15 minutes. This being a RU-vid video, you can pause it or put it on 75% speed, if it helps. I'm assuming my audience is working through Real Analysis while watching my videos, like my class was when I was teaching it. These are meant as a supplement to help guide people through a difficult topic.

@victorsauvage1890 5 месяцев назад

Try sitting on your hands when you offer an explanation. Are you serious? Is this a comedy show? No beginner can follow your wild hand gestures. You seem to be portraying an absent-minded professor.

@JoelRosenfeld 5 месяцев назад

I’m sorry you don’t like the videos. This video was made to augment my intermediate analysis class and many others have told me they find these videos helpful. What’s your background? How do you think it could be made better?