How to prove that an integer exists matching the criteria I am working over the review for my next exam coming up in Calculus, and am stuck on this problem:

Prove that, if $\lim_{n \to \infty}a_n=a$ and $a>0$, then there exists a positive integer $N$ such that $a_n>0$ for all $n>N$.

So, first I thought of using the property of a limit, that is, let $\epsilon > 0$, because the sequence $\{a_n\}$ converges to $a$, we are able to write $|a_n-a|<\epsilon$. However, I'm not sure where to go from there.
Now, the problem seems to want a proof using the Archimedean Property, however I am unsure of how exactly to begin applying it, maybe assume that $|a_n-a|\geq 0$, and thus, I could find an integer $N$ such that, for all $n>N$, $0 < \frac{1}{n} < |a_n-a|$. However, I don't see how to get there.

Since $\,a>0\,$ there exists $\,\epsilon>0\,\,\,s.t.\,\,\,a-\epsilon>0\,$ . Now, there exists 
$$N_\epsilon\in\Bbb N\,\,\,s.t.\,\,\,n>N_\epsilon\Longrightarrow |a_n-a|<\epsilon$$
But 
$$|a_n-a|<\epsilon\Longleftrightarrow a-\epsilon<a_n<a+\epsilon$$
Well, wrap the above up and end the argument.
By the way, you can choose $\,\epsilon=a/2\,$, say.
