# Prove that a converging sequence is bounded

1. Aug 31, 2008

### 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?

2. Aug 31, 2008

### Hitman2-2

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

3. Aug 31, 2008

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 ...

Last edited by a moderator: Aug 31, 2008
4. Aug 31, 2008

### Dick

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