Contraction Mapping Theorem on a Nonempty Complete Metric Space

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $(X,d)$ be a nonempty complete metric space and let $f:X\to X$ be a contraction.

(i) There is exactly one $p\in X$ with $f(p)=p$.

(ii) For every sequence $(x_{n})$ in $X$ with $x_{n+1}=f(x_{n})$ for every $n\in\mathbb{N}$, and every $p\in X$ with $f(p)=p$, $(x_{n})$ converges to $p$ in $(X,d)$.