Prove that a converging sequence is bounded

[kehler]

Homework Statement

Suppose that the sequence {a_n}converges. Show that the sequence {a_n} 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,
|a_n - x|< delta. Therefore, for n>=N, -delta+x < a_n < 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, a_n 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, a_n 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).