Let $f:[3,\infty)\to \mathbb R$ defined as $x\mapsto \sqrt x$. Then for $ x,y \in [3,\infty)$ we have $$ \left|\sqrt x-\sqrt y \right| =\left|\frac{x- y}{\sqrt x+\sqrt y } ~\right|\leq \frac 1 {2\sqrt3}\left|x-y\right|.$$
Hence $f$ is contraction on the complete space $[3,\infty)$. But clearly it has no fix point in $[3,\infty)$. What I am missing here?