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Let $(X,d)$ be a metric space. Let $f: X \to X$ be a map such that there exists $C \in [0,1/2)$ and satisfy $d(f(x),f(y)) \leq C[d(x,f(x))+d(y,f(y))]$ for all $x,y \in X$.

I want to prove that if $f$ has a fixed point then $X$ is complete.

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  • $\begingroup$ some $f$ or any $f$? $\endgroup$ – Jens Renders Jul 23 '17 at 12:29
  • $\begingroup$ for every $f$ satisfying this condition $\endgroup$ – Manu Rohilla Jul 23 '17 at 12:30
  • $\begingroup$ i.e. every map of this kind has a fixed point $\endgroup$ – Manu Rohilla Jul 23 '17 at 12:31
  • $\begingroup$ Not clear for me what you want to prove: If every $f$ satisfyinfg this condition has an extreme point then $X$ is complete? or if some $f$ satisfies this and has an extreme point then $X$ is complete? $\endgroup$ – John D Jul 24 '17 at 13:53
  • $\begingroup$ if every $f$ satisfying this condition has a fixed point then $X$ is complete. $\endgroup$ – Manu Rohilla Jul 24 '17 at 15:51

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