Help with Intro to Real Analysis proof with sequence and limit.

**Problem states:**

Let http://www.cramster.com/Answer-Board...9375001985.gif be a sequence in http://www.cramster.com/Answer-Board...8125006971.gif with http://www.cramster.com/Answer-Board...9062507335.gif http://www.cramster.com/Answer-Board...4375007946.gif= a.

If a > 0, prove http://www.cramster.com/Answer-Board...0937504331.gif (the set of Natural numbers) such that http://www.cramster.com/Answer-Board...6875002318.gif.

**My work so far:**

I don't have a proof written up for this problem yet, but I have jotted down some ideas that I think might help me:

I know that the sequence {$\displaystyle a_n$} converges to a and a > 0.

So by definition of convergence, I know that there exists an $\displaystyle n_0$ in the Naturals such that

|$\displaystyle a_n$ - a| < ε for all n ≥ $\displaystyle n_0$.

There is also the theorem that every convergent sequence is bounded, so by definition of bounded: there exists an M > 0 such that

|$\displaystyle a_n$| ≤ M for all n in the Naturals.

So for this proof do I need cases? Would I do proof by contradiction?

I worked with a study group on this, and my classmate drew a picture and talked us through the thinking. It kind of made sense to me that if a sequence converges to zero, then eventually the sequence has to be greater than zero at some point.

Any help, suggestions, and/or tips are greatly appreciated.

Thank you for your time! =)