# help! show this sequence is Cauchy

• Feb 16th 2011, 03:27 PM
mremwo
help! 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 PM
Plato
Quote:

Originally Posted by mremwo
Suppose that 0<r<1 and $\displaystyle \ \mid x_{n+1} - x_{n}\mid \ \forall n \in N$ Show that $\displaystyle (x_{n})$ is Cauchy.

There is a great deal missing here.
In fact, there is no question there what so ever.
• Feb 16th 2011, 03:33 PM
mremwo
yes i just edited it! sorry!
• Feb 16th 2011, 04:42 PM
DrSteve
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 PM
mremwo
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!