Does the series converge or diverge?

[╔(σ_σ)╝]

Homework Statement

Let $$x_{n}$$ be a monotone increasing sequence such that $$x_{n+1}-x_{n}< \frac{1}{n}.$$ Must $$x_{n}$$ converge ?

Homework Equations

Instinctively, I think it converges since the terms "bunch" up as n increases.

The Attempt at a Solution

$$|x_{n+1}-L| \leq |x_{n+1}-x_{n}| + |x_{n}-L|$$.
$$|x_{n+1}-L|< \frac{1}{n}+ |x_{n}-L|$$ . But this doesn't tell me anything about convergence.

If I apply the triangle inequality continuosly all I can see is that
$$|x_{n+1}-L|< \sum\frac{1}{n}$$

What can you guys tell me ?

 

[╔(σ_σ)╝]

Great that makes sense. I had a feeling it need not converge but I was unsure. Does this work too...
Define a sequence such that $$x_{n}-x_{n-1}= \frac{1}{n}$$ such that the sequence has positive terms them
$$|x_{n} -L| =|1 + 1/2 + ...+ 1/n|[/tex ] which does not converge.<br /> I suppose that also works correct? <br /> Excuse my latex, typing from a phone is difficult.$$

 

[vela]

Yes, that works.

 

[╔(σ_σ)╝]

Thanks a lot for the help.

 

[gomunkul51]

So what have you decided, does it converges or diverges? :)