Francis Su is a professor at Harvey Mudd College and is a renowned math educator. He taught a 26-lecture real analysis class at Harvey Mudd in 2010 and video of his lectures are all online. This is a great resource if you want high-quality, lecture-style instruction of any topic, or of an entire course. Above is Lecture 1, and if you go to this page on YouTube you'll see all 26 lectures threaded.

Jason Bramburger is a professor at Concordia University. I will admit that I do not know as much about him, but in 2022 he recorded his 30-lecture real analysis class and I have heard it is well done. Above is Lecture 1, and if you go to this page on YouTube you will see all 30 lectures threaded.

In this video, Vi Hart proves the Pythagorean theorem using paper folding.

Mathologer shows some fun visualizations of the Pythagorean theorem, which we discussed in Section 1.3 as the preamble towards the first big indication that the rationals simply don't cut it, and to properly do mathematics we need the reals.

In this video, Vi Hart imagines the reals as an infinite tree, and sees what that would imply.

The first proof of the book, in Example 1.4, was that the square root of 2 is irrational. Importantly, π is also irrational, and this video discusses why.Check out this video (and __this one__!) especially if you haven't seen continued fractions before. It's pretty amazing to see how simple some special numbers' continued fractions can be---even special numbers which are irrational or even transcendental.

In Chapter 1 we discuss the essential properties of the real numbers and Theorem 1.20 says that the reals really do exist and that they are the unique complete ordered field. I then casually mentioned that in Appendix A I sketch a way to construct the real numbers. In case you prefer watching to reading, this video from PBS Infinite Series also sketches how this construction is done.

In Section 2.2 we talked about the amazing fact that there is more than one size of infinity. Since cardinality is defined in terms of bijections, it all comes back to that fundamental definition when determining whether two infinite sets are the same size or not. This video discusses these interesting ideas and their remarkable consequence.

Also check out Vi Hart's take on this hierarchy---and what different infinites feel like.

Corollary 2.14 states that there are infinitely many infinites. This was shown by first noting that the power set of a set *A* has a cardinality strichtly larger than |*A*|, even if *A* is an infinite set. And so by taking successive power sets, we get a growing list of infinites. But if there are infinitely many infinities, and infinites come in different sizes... then which infinity tells you how many infinities there are? This video from PBS Infinite Series discusses this question.

Here's one final video from PBS on infinity. Instead of focusing on cardinals, their focus turns to ordinals. (But if you're pining for even more, also see __this video__ from them.)

This video deals with a sequence which is quite easy to define, but remarkably difficult to solve. It is one of the hardest problems that I slipped in as an open question in the book---this one being the first open question of Chapter 3.

Fun with series of shapes! From __Apollonian gaskets__ to elephants to camels, and more.

A enlightening video on one of the most mysterious results in math.

Proposition 4.15 in the book shows that the harmonic series diverges. This and more, plus scenes from *The Simpsons*, are in this Mathologer video.

In the book's largest footnote, which took up nearly half of page 133, I mentioned that the alternating harmonic series converges to ln(2). What about the alternating sum of just inverse *odd* numbers? Then the answer has to do with π! And if there's a π, then somewhere there's a circle. But where is it in odd numbers? This video finds a connection, which ties into prime numbers. This video takes more concentration than most of the others on this page, but like all 3Blue1Brown videos, it's worth it.

Theorem 4.23 in the book is one of my favorite theorems. It says that you can rearrange a conditionally convergent series and have it converge to... anything you want. Here's a video from Mathologer that discusses this and more.

In Chapter 5 we study the topology of the real numbers, but one can also study the topology of general sets, where the open, closed and compact properties likewise generalize. Topology also classifies shapes, and at times considers shapes the same if one can one continuously deform one into the other (cue a joke about coffee mugs and doughnuts). Other shapes like Möbius strips are notable for their unique properties. Here's a video where this topology comes up naturally.

Want another 10 minute introduction to the broader field of topology? Check out this video from PBS Infinite Series.

Find someone who looks at you the way this guy looks at a Klein bottle. That is all.

Example B.2 of Appendix B discusses a theorem which implies that there are two antipodal points on the Earth which have *exactly* the same temperature. This video from 3Blue1Brown discusses other connections to this result, including a discrete version in which a pair of thieves are trying to evenly split their stolen necklace.

Example B.7 of Appendix B deals with how to find stable footing for a good table situated on uneven ground. This video also addresses this problem, plus adds an additional application to so-called *hugging squares*. They are two fun applications of the intermediate value theorem (Theorem 6.38).

Space-filling curves are continuous (two-dimensional) curves which somehow manage to completely fill up three-dimensional space. A famous such curve is called the Hilbert curve and is discussed in Example B.14 of Appendix B. A real-world application of this curve to videography and audiology is discussed in this video.

Vi Hart's introduction to the derivative by explaining what the derivative is for a car ride---and then what the second and third derivatives correspond to.

3Blue1Brown has an 12-part video series on the "Essence of Calculus." The derivative is discussed from many different perspectives. The rest of the videos relating to derivatives are below.

Here's the second derivative video in the aforementioned series.

Here's the third derivative video in the aforementioned series. It's on means to visualize the chain rule (Theorem 7.13) and the product rule (Theorem 7.11).

Here's the fourth derivative video in the aforementioned series. Among other things, it's on l'Hopital's Rule (Theorem 7.27).

Here's the final derivative video in the aforementioned series.

The open question in the derivative chapter (page 256) asked about why if take the derivative of the π*r*² formula for the area of circle, do you get the formula for the diameter: 2π*r*. And, more importantly, how deep does this correspondence go? Related issues between geometric formulas are discussed in this video.

The devil's staircase is a function which has a derivative of zero nearly everywhere, and yet somehow climbs from (0,0) to (1,1). And here's the crazy thing: It does so *continuously*. This is the topic of Example B.8 in Appendix B, as well as of this video by PBS Infinite Series.

This video, from Numberphile, discusses infinitesmals, which were originally used to define the integral, and prove theorems thereof. This video also touches on some of the interesting history of this development, including the rivalry between Newton and Leibniz.

A great video on FTC.

A video connecting slopes to areas, and hence derivatives to integrals.{

Section 8.8 was about the *measure zero integrability criterion,* and this idea of measure was investigated further in Example B.17 of Appendix B. This idea of measure is important in real analysis and a connection between it and music is discussed in this 3Blue1Brown video.

The final open questions of Chapter 2 (on page 63) deal with the *axiom of choice*. This video talks generally about this axiom, as well as a specific application to the measure of sets, which was discussed in the zero case in Section 8.8. This specific application was also discussed in Fact B.17 of Appendix B.

This is the excellent video on Taylor series that I kept pestering you to watch in Section 9.9. Given an infinitely differentiable function *f*, this video describes how to construct polynomials that approach *f*.

This short video gives a good way to think about why *e^*iπ equals -1. It of course involves imaginary and so leaves the realm of *real* analysis, but not in any way that you can't handle. A second video on this topic by Mathologer is __here__.This equation was deduced in the final section of the book as a consequence of Taylor series.

This equation was deduced in the final section of the book as a consequence of Taylor series.

The Riemann Hypothesis is one of the great unsolved problems in mathematics. It was mentioned on page 53 of the text, but not stated or discussed. This is an excellent video discussing it.

The Heat Equation is currently an important and active research topic in analysis. This video is a great introduction to this field.

Check out this video for a surprising appearance of π in a physics problem. Throughout the discussion you'll notice several tools discussed in real analysis.

A number is *algebraic* if it is the root of some polynomial with integer coefficients, and it is *transcendental* if it is not. Showing a number is irrational is one thing, but showing it is transcendental is often much more difficult. This video discusses how one might try to prove that *e* and π are transcendental.

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