Theorem 3 (Weierstrass-Stone)

Let X be a compact space and let H be a set of continuous real-valued functions on X which separates the points of X. Then every continuous real-valued function on X can be uniformly approximated by polynomials (with real coefficients) in the functions of H.

An equivalent statement of the theorem is that any subalgebra of $\mathscr{C}(X; \mathbf{R})$ which contains the constant functions and separates the points of X is dense in $\mathscr{C}(X; \mathbf{R})$.

Let $H_0$ be the set of all polynomials in the functions of H, and let $\overline{H}_0$ be the closure of $H_0$ in $\mathscr{C}$. If g is any polynomial in n variables with real coefficients, then $(u_1, u_2, \ldots, u_n) \to g(u_1, u_2, \ldots, u_n)$ is a continuous mapping of $\mathscr{C}^n$ into $\mathscr{C}$, which maps $H_0^n$ into $H_0$, and therefore maps $\overline{H}_0^n$ into $\overline{H}_0$ (Chapter I, § 2, no. 1, Theorem 1). In particular, $\overline{H}_0$ is a vector subspace of $\mathscr{C}$ and evidently satisfies the first and third conditions of Theorem 2; we shall show that it also satisfies the second condition, and this will prove that $\overline{H}_0 = \mathscr{C}$.

Since every function $u \in \overline{H}_0$ is bounded in $X$, it is enough to prove the following lemma: