If $a_n$ and $b_n$ are converging sequences, and for all $n\in\mathbb Z+$, $a_n \leq b_n$, then $\lim_{n\to\infty} a_n \leq \lim_{n\to\infty} b_n$.
Given $a_n, b_n$ are both convergent sequences such that $a_n \leq b_n$ for all $n$, show $\lim_{n\to\infty} a_n \leq \lim_{n\to\infty} b_n$.