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 $\displaystyle lim_{n\rightarrow \infty}\{a_n-b_n\}=a-b$ and that $\displaystyle lim_{n\rightarrow \infty}\{a_n-b_n\}\leq0$ since $\displaystyle a_n\leq b_n$, but I'm stuck on proving it. Any help appreciated. Thanks!

Here is the problem:

Suppose that $\displaystyle \{a_n\}$ and $\displaystyle \{b_n\}$ are both Cauchy sequences and that $\displaystyle a_n\leq b_n$ for each n. Prove that $\displaystyle lim_{n\rightarrow\infty}a_n\leq lim_{n\rightarrow\infty}b_n$