Showing that a_n > 0 given its decreasing and lim a_n=0

[Mr Davis 97]

I'm reading a proof, and it says that since $\lim_{n\to\infty}a_n = 0$ and $a_{n+1}\le a_n$ for all $n$, it must be the case that $a_n \ge 0$.

This seems very obvious, since if $a_n < 0$ for some $n$ then it would decreasing from then on, so there wouldn't be infinitely many elements contained in every $\epsilon$-neighborhood around $0$. But is this sufficient for proof? What would a write up of a "proof" of this simple statement look like?

 

[andrewkirk]

Assume the opposite. Then there's some $a_n<0$. Then show that every subsequent member of the series is at least distance $|a_n|$ from 0. So 0 can't be the limit.