Problem states:

Let be a sequence in with = a.

If a > 0, prove (the set of Natural numbers) such that .

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 { } converges to a and a > 0.

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

| - a| < ε for all n ≥ .

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

| | ≤ 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! =)