Prove that a converging sequence is bounded

[kehler]

1. The problem statement, all variables and given/known data
Suppose that the sequence {an}converges. Show that the sequence {an} is bounded.

3. The attempt at a solution
Since the sequence converges, for every delta>0, there must exist a number N such that for every n>=N,
|an - x|< delta. Therefore, for n>=N, -delta+x < an < delta + x.
So I've proven that for n>=N, the sequence is bounded between -delta+x and delta+x.

But I don't know how to prove that for n<N, an is also bounded. I know that there are only a finite number of elements before the sequence starts to converge. Is there a theorem stating that all finite sets are bounded?

[Hitman2-2]

But I don't know how to prove that for n<N, an is also bounded. I know that there are only a finite number of elements before the sequence starts to converge. Is there a theorem stating that all finite sets are bounded?

I would consider looking at the maximum of a finite set.

[statdad]

To expand a little:
Pick an \varepsilon > 0 , and use convergence to conclude that


|x_i - a | < \varepsilon


for all i \ge N .

You now have two sets of elements of the sequence: those with i \ge N and those for smaller i . You should be able to argue that both sets are bounded, which means ...

[Dick]

Yes. All finite sets are bounded. Prove it by induction. (If you really need a proof).