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.

Hint: Let $\epsilon=\dfrac{a}{2}$. By the definition of limit, there is an $n$ such that if $n\gt N$ then $|a_n-a|\lt \epsilon$.  Conclude from this that if $n\gt N$ then $a_n\gt 0$. 
