# Proving divergence of a sequence

1. Sep 22, 2013

### dimanet

Hello!

Suppose that a sequence $\{s_n\}$ of positive numbers satisfies the condition $s_{n+1} > \alpha s_n$ for all $n$ where $\alpha>1.$ Show that $s_n \to \infty.$

I can't prove this using definition and given condition, I can only give an example of such sequence, $\exp n$ with $\alpha = 2$ but it seems useless for me. The only I know is that sequence is monotone and unbounded and so diverges.

I'm only asking about a little hint that will give me a possibility to solve this by myself.

Thanks a lot.

2. Sep 22, 2013

### UltrafastPED

Hint: Does this sequence have a limit point?

3. Sep 22, 2013

### dimanet

No. This sequence doesn't have a limit point but infinity. Right? But anyway I still don't understand. Let me to think some time. :-)
Thanks.

4. Sep 22, 2013

### mathman

The condition implies sn > αns0 ->∞ as long as s0 > 0.

5. Sep 22, 2013

### dimanet

Aha, now I understand the logic behind a given condition. Thank you!