Contraction Mapping Theorem on a Nonempty Complete Metric Space

Let $(X,d)$ be a complete metric space, and suppose that $X$ is nonempty. Let $T:X\to X$ be a contraction. Then $T$ has a unique fixed point in $X$.

Moreover, for every $x_0\in X$, the iterated sequence

converges to that fixed point, where $\mathbb{N}$ denotes the natural numbers.