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[SQUEAKING] [RUSTLING] [CLICKING] CASEY RODRIGUEZ: All right, so we proved these two theorems last time, and we used them for-- and we had a couple of applications of them. So the first theorem, simple theorem, was that a sequence converges to x if and only if the limit as n goes to infinity of the absolute value of xn minus x goes to 0. And then we also had the squeeze theorem, that if you have two-- if you have three sequences-- a sub n, b sub n, and x sub n-- so that x sub n is in between a sub n and b sub n, and the a sub n and b sub n converge to the same thing, then the sequence x sub n converges, and it converges to the [? common ?] limit of a sub n and b sub n. So x sub n gets squeezed in between a sub n and b sub n. Last time we used this to prove a kind of special limit that this equals 0 if the absolute value of c is less than 1. So I had proved it for c positive, and maybe c less than 1-- but same proof works with absolute values, just because the absolute val...

[SQUEAKING] [RUSTLING] [CLICKING] CASEY RODRIGUEZ: All right. So last lecture, we concluded our discussion about measurable functions, I mean, measurable sets. And remember our original motivation was that we're trying to build an integral that somehow surpasses the properties of that of the Riemann integral in that, hopefully, this larger class of functions that are integral with respect to this integral form a Banach space as opposed to Riemann integral functions. And we started off by asking the question, how would we integrate the simplest types of functions, which are just one on a set and zero off of it. And that led us to how to define measure. And so then we define Lebesgue measurable sets, proved that they form this special type of collection of sets called the sigma algebra, and that they contain a lot of sets, a lot of interesting sets, open, closed, unions, countable unions of closed, which are not necessarily closed, countable intersections of opens, whic...

[SQUEAKING] [RUSTLING] [CLICKING] CASEY RODRIGUES: So I'm going to prove a few theorems about limits, which will allow us to compute limits, or at least we can use to prove that other non-trivial limits exist using these theorems, rather than using the definition directly. So this first theorem is the easiest theorem in the world because it's simply just restating the definition of convergence of a sequence. So I'm going to state it as follows, so pretty short, that if I have a sequence x sub n, then it converges to x if and only if the sequence obtained by taking the absolute value of x of n minus x equals 0, or the limit of that sequence is 0. So what is the proof? It follows just immediately from the definition. So I'm not even going to write anything. I'll leave it to you. But the proof follows from the definition and the simple fact that x sub n minus x in absolute value is equal to the absolute value of the absolute value of x sub n minus x minus...