Existence of fixed point implies completeness

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.

• some $f$ or any $f$? – Jens Renders Jul 23 '17 at 12:29
• for every $f$ satisfying this condition – Manu Rohilla Jul 23 '17 at 12:30
• i.e. every map of this kind has a fixed point – Manu Rohilla Jul 23 '17 at 12:31
• 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? – John D Jul 24 '17 at 13:53
• if every $f$ satisfying this condition has a fixed point then $X$ is complete. – Manu Rohilla Jul 24 '17 at 15:51