Lecture 7: Bolzano–Weierstrass Theorem; Cauchy Sequences; Series

The Bolzano-Weierstrass theorem says that any bounded sequence has a convergent subsequence. This crucial fact can be used to show other important theorems. An almost immediate consequence of it is the Cauchy convergence theorem. We also introduce the notion of a series and get acquainted with perhaps the single most important series: the geometric series. For series, the most important question is whether or not it is convergent. To determine that, there are a number of tests beginning with the two versions of the comparison test.