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