Lipschitz Extension of a Real Function from a Subset of a Metric Space

In the setting of Separable Real Normed Spaces and Hahn–Banach: Notation, let $(X,d)$ be a metric space and let $D\subseteq X$ be nonempty.

(i) Lipschitz extension. Let $L\ge0$ be a real number and let $f:D\to\mathbb{R}$ be a function with

$$|f(u)-f(v)|\le L\,d(u,v)\qquad\text{for all }u,v\in D.$$

Then there is a function $F:X\to\mathbb{R}$ with $F(u)=f(u)$ for every $u\in D$ and

$$|F(x)-F(y)|\le L\,d(x,y)\qquad\text{for all }x,y\in X.$$