Existence of fixed point implies completeness

Question

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.