Proving divergence of a sequence

[dimanet]

Hello!
Please help me to solve following exercise (2.5.8) from Elementary Real Analysis by Thomson-Bruckner:

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.

 

[UltrafastPED]

Hint: Does this sequence have a limit point?

 

[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.

 

[mathman]

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

 

[dimanet]

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