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## Homework Statement

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

## Homework Equations

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

## The Attempt at a Solution

[tex] |x_{n+1}-L| \leq |x_{n+1}-x_{n}| + |x_{n}-L|[/tex].

[tex] |x_{n+1}-L|< \frac{1}{n}+ |x_{n}-L|[/tex] . But this doesn't tell me anything about convergence.

If I apply the triangle inequality continuosly all I can see is that

[tex] |x_{n+1}-L|< \sum\frac{1}{n}[/tex]

What can you guys tell me ?