So I teach high school math, but I love to study math in my spare time. I'm charging my way through an analysis book, trying to solve all of the problems. Been looking at this one for a couple of days now and I need to get un-stuck. It's clear to me that and that since , but I'm stuck on proving it. Any help appreciated. Thanks!
Here is the problem:
Suppose that and are both Cauchy sequences and that for each n. Prove that
is a snake that crawls arbitrarily close to . On the other hand, if , then there is a finite distance between and 0. Eventually the distance between and becomes smaller than this distance. What happens then?