Lecture 8 The Squeeze Theorem and Operations Involving Convergent Sequences
[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...