Exercise 10, § 4

Use the Weierstrass-Stone theorem to give a new proof of Urysohn’s theorem (Chapter IX, § 4, no. 2, Theorem 2) for closed subspaces of a compact space $X$. [If $F$ is a closed subspace of $X$, consider the set $H$ of restrictions to $F$ of continuous mappings $X \to \mathbf{R}$, and observe that if $f \in H$ and $|f(x)| \leq a$ for all $x \in F$, then there exists a function $g \in H$ which coincides with $f$ on $F$ and is such that $|g(x)| \leq a$ in $X$.]