November 10th, 2016, 03:49 PM  #1 
Member Joined: Nov 2016 From: Kansas Posts: 44 Thanks: 0  Cauchy Sequence II
How to solve this?

November 10th, 2016, 05:42 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,541 Thanks: 2146 Math Focus: Mainly analysis and algebra 
The obvious approach would be to derive the closed form for the sum $$s_n = \frac{1x_0^{n+1}}{1x_0}$$and then show that $s_n$ is Cauchy. I would begin by noting that for any given $N$, we have $$s_N \lt s_n \lt \frac1{1x_0} \quad \text{for all $n \gt N$}$$ Thus, for any given $\epsilon$, we can pick $N$ such that $\frac1{1x_0}s_N \lt \epsilon$. Note that the above only works for $0 \le x_0 \lt 1$, but the approach can be easily modified for $1 \lt x_0 \lt 0$. Last edited by v8archie; November 10th, 2016 at 05:50 PM. 
November 11th, 2016, 03:06 PM  #3  
Global Moderator Joined: May 2007 Posts: 6,214 Thanks: 491  Quote:
 

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