- #1
Damidami
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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!