Suppose that 0<r<1 and $\displaystyle \ \mid x_{n+1} - x_{n} \mid \ $ < $\displaystyle r^n \ \forall n \in N$

Show that $\displaystyle (x_{n})$ is Cauchy.

Thank you!

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- Feb 16th 2011, 03:27 PMmremwohelp! show this sequence is Cauchy
Suppose that 0<r<1 and $\displaystyle \ \mid x_{n+1} - x_{n} \mid \ $ < $\displaystyle r^n \ \forall n \in N$

Show that $\displaystyle (x_{n})$ is Cauchy.

Thank you! - Feb 16th 2011, 03:32 PMPlato
- Feb 16th 2011, 03:33 PMmremwo
yes i just edited it! sorry!

- Feb 16th 2011, 04:42 PMDrSteve
Some hints:

Use the generalized triangle inequality.

$\displaystyle \sum r^n$ is a convergent geometric series, thus the "tail" of the series can be made arbitrarily small. - Feb 16th 2011, 06:40 PMmremwo
I'm sorry, I still have no idea what to do. Could you give me another hint? This concept is very hard for me to grasp

EDIT: nevermind, i got it! thank you!