Proposition 3

Let $(K_i)_{i \in I}$ be a family of compact intervals of $\mathbf{R}$, $K = \prod_{i \in I} K_i$ their product, and let $X$ be a compact subspace of $K$. Then every continuous real-valued function on $X$ can be uniformly approximated by polynomials in the coordinates $x_i = \mathrm{pr}_i x$.

For if $x = (x_i)$ and $y = (y_i)$ are two distinct points of $X$, there is at least one index $i$ such that $x_i \neq y_i$; hence the family of continuous functions $\mathrm{pr}_i$ satisfies the conditions of Theorem 3.