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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 RODRIGUEZ: OK, so last time, we were talking about sequences, and I introduced the notion of a limit of a sequence. So we say that x n converges to x if for all epsilon positive there exists an M, a natural number, such that for all little n bigger than or equal to capital M we have x sub n minus x is less than epsilon. So what this definition says is that given any little bit of tolerance, as long as I go far enough out in the sequence, the entries are getting within that tolerance to x. And when x then converges to x, we'll write x is equal to the limit as n goes to infinity of x sub n, or using this notation, x sub n arrow x. And so I've said before that if you ever have a reasonably complicated or interesting definition, you should try to come up with examples and negate it. So the end of last time, we saw a few examples of-- at least one example of-- a convergent sequence. Let's negate this definition because then w...