Proof of McShane Extension of a Real-Valued Lipschitz Function on a Metric Space

We use the notation of the statement. By Metric Space the metric $d$ satisfies $0\le d(u,v)$, $d(u,u)=0$, $d(u,v)=d(v,u)$ and $d(u,w)\le d(u,v)+d(v,w)$ for all $u,v,w\in X$.

Claim 1. Fix $x\in X$. Since $A$ is nonempty we may fix $a_{0}\in A$, and then $f(a_{0})+L\,d(x,a_{0})\in S(x)$, so $S(x)$ is nonempty.

We show that $f(a_{0})-L\,d(x,a_{0})$ is a lower bound for $S(x)$. Let $a\in A$. By claim 3 of Properties of the Absolute Value in an Ordered Field and the Lipschitz hypothesis,

The triangle inequality gives $d(a_{0},a)\le d(a_{0},x)+d(x,a)=d(x,a_{0})+d(x,a)$, and multiplying this inequality by the nonnegative number $L$ preserves it, so

As $a\in A$ was arbitrary, $f(a_{0})-L\,d(x,a_{0})$ is a lower bound for $S(x)$. Being nonempty and bounded below, $S(x)$ has a greatest lower bound in $\mathbb{R}$ by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below, so $F(x)=\inf S(x)$ is a well-defined real number.

Claim 2. Let $a\in A$. Taking the point $a$ itself in the definition of $S(a)$ and using $d(a,a)=0$ shows $f(a)=f(a)+L\,d(a,a)\in S(a)$, whence $F(a)\le f(a)$. Applying the lower bound established in claim 1 with $x=a$ and $a_{0}=a$ shows that $f(a)-L\,d(a,a)=f(a)$ is a lower bound for $S(a)$; since $F(a)$ is the greatest lower bound, $f(a)\le F(a)$. Therefore $F(a)=f(a)$.

Claim 3. Let $x,y\in X$ and let $a\in A$. By the triangle inequality $d(x,a)\le d(x,y)+d(y,a)$, and multiplying by $L\ge0$ and adding $f(a)$ gives

the first inequality because $f(a)+L\,d(x,a)\in S(x)$ and $F(x)$ is a lower bound for $S(x)$. Hence

so $F(x)-L\,d(x,y)$ is a lower bound for $S(y)$ and therefore $F(x)-L\,d(x,y)\le F(y)$, that is $F(x)-F(y)\le L\,d(x,y)$. Exchanging the roles of $x$ and $y$ and using $d(y,x)=d(x,y)$ gives $F(y)-F(x)\le L\,d(x,y)$. By claim 6 of Properties of the Absolute Value in an Ordered Field the two inequalities together give $|F(x)-F(y)|\le L\,d(x,y)$, so $F$ is Lipschitz with constant $L$ in the sense of Lipschitz Map Between Metric Spaces.