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. On the left is lecture 1, but if you go to this page on Youtube you'll see all 26 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.
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.
On page 129 in the series chapter, the Basel problem is discussed. But what does the sum of inverse squares have to do with π? For centuries this connection was unclear, but a 2010 paper provided new insights. In this 3Blue1Brown video, this connection is beautifully visualized as measuring light from a growing collection of circularly arranged lighthouses.
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.