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## Homework Statement

The problem is longer but the part I'm stuck is to show that [itex] \{x_n\} [/itex] is convergent (I thought showing it is Cauchy) if I know that for all [itex] \epsilon > 0 [/itex] exists [itex] n_0 [/itex] such that for all [itex] n \geq n_0 [/itex] I have that

[itex] |x_{n+1} - x_n| < \epsilon[/itex]

## Homework Equations

A sequence is Cauchy if for all [itex] \epsilon > 0 [/itex] and for all [itex] n,m \geq n_0 [/itex] one has

[itex] |x_m - x_n| < \epsilon [/itex]

## The Attempt at a Solution

I called [itex] m = n+p [/itex] (for [itex] p [/itex] an arbitrary positive integer)

Then

[itex] |x_m - x_n| = |x_{n+p} - x_n|[/itex]

But (and I think there is some mistake here):

[itex] |x_{n+1} - x_n| < \epsilon/p [/itex]

[itex] |x_{n+2} - x_{n+1}| < \epsilon/p [/itex]

[itex] \vdots [/itex]

[itex] |x_{n+p} - x_{n+p-1}| < \epsilon/p [/itex]

So

[itex] |x_{n+p} - x_n| < \underbrace{|x_{n+1} - x_n|}_{< \epsilon/p} + \underbrace{|x_{n+2} - x_{n+1}|}_{< \epsilon/p} + \ldots + \underbrace{|x_{n+p} - x_{n+p-1}|}_{< \epsilon/p} < \epsilon [/itex]

Any help on why it's wrong (if it is) and how to solve it correctly?

Thanks!