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Let $(X, d)$ be a complete metric space, $r∈ (0,1)$ and $\{x_n\}$ be a sequence in $X$ such that

$$d(x_{n+2}, x_{n+1})≤ rd(x_{n+1}, x_n),$$ for every $n∈ℕ$. Show that $\{x_n\}$ is a convergent sequence.

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Out of curiosity, where did you learn LaTeX? –  JavaMan Dec 10 '12 at 5:09
    
@JavaMan: It seems to me that we’ve been seeing more of this half-and-half coding lately. –  Brian M. Scott Dec 10 '12 at 5:21
    
@Brian: I agree. Michael Hardy even brought it up in Meta. I'm curious to find out from where it originates. –  JavaMan Dec 10 '12 at 6:55
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1 Answer

HINT: Show that the sequence is Cauchy. What can you say about $\sum_{k\ge 0}d(x_{n+k},x_{n+k+1})$? You’ll be using the triangle inequality and looking at the sum of a geometric series.

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