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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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. Please edit your post.
yes i just edited it! sorry!
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.
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!
Last edited by mremwo; Feb 16th 2011 at 09:30 PM.
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