Let $(X,d)$ be a \reftext{def:complete-metric-space-2026a}{complete metric space}, and suppose that $X$ is nonempty. Let $T:X\to X$ be a \reftext{def:contraction-metric-space-2026a}{contraction}. Then $T$ has a unique \reftext{def:fixed-point-self-map-2026a}{fixed point} in $X$.

Moreover, for every $x_0\in X$, the iterated \reftext{def:sequence-in-set-2026a}{sequence}

\reftext{def:convergent-sequence-metric-space-2026a}{converges} to that fixed point, where $\mathbb{N}$ denotes the \reftext{def:natural-numbers-2026a}{natural numbers}.