Hi Scholars!
Recently, we have been focusing on topics around sequences. So, this weekend I made a graphic to illustrate one way to prove the Bolzano Weierstrass Theorem, which states each bounded sequence of real numbers has a convergent subsequence. It was actually a bit more involved than I expected, but I don’t think I will be able to forget the core ideas now. The proof breaks down into two cases, whether or not there are an infinite number of “peaks.” A “peak” is where a point in the sequence is at least as big as all future points in the sequence. If there are infinitely many peaks, then you can make a subsequence of successive peaks. If there are not infinitely many peaks, then after you pass the final peak in the sequence you can always find a point that is bigger, meaning you can generate an increasing subsequence. Either way, you are able to construct a monotone subsequence. And, because you assumed it was bounded, the monotone convergence theorem can be applied to determine the subsequence converges.
Stay Awesome.
Howard