# Prove that a converging sequence is bounded

## Homework Statement

Suppose that the sequence {an}converges. Show that the sequence {an} is bounded.

## 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?

## Answers and Replies

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
Homework Helper
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 ...

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Dick
Science Advisor
Homework Helper
Yes. All finite sets are bounded. Prove it by induction. (If you really need a proof).