Exercise 10, § 4

Deduce from the Weierstrass-Stone theorem a new proof of the Urysohn theorem (IX, p. 44, Theorem 2) for the closed subspaces of a compact space X. (If F is closed in X, consider the set H of the restrictions to F of the continuous mappings of X into R, and observe that if $f \in H$ and $|f(x)| \leq a$ in $F$, there exists a function $g$ which coincides with $f$ in $F$ and is such that $|g(x)| \leq a$ in $X$.