Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 22F / b

For each positive integer $i$, apply the Stone-Weierstrass theorem on the closed ball $\overline{B}(0,i)$ to choose a polynomial $p_i$ in $n$ variables satisfyingEvery compact $B\subset {\mathbb{R}}^n$ lies in some such ball, so the same estimate for all sufficiently large $i$ proves that $p_i{|}_B\to f{|}_B$ uniformly.

Now suppose instead that $p_i\to f$ uniformly on all of ${\mathbb{R}}^n$. For sufficiently large $i,j$, the polynomial $p_i-p_j$ is bounded on ${\mathbb{R}}^n$, and every nonconstant real polynomial is unbounded along a suitable line. Hence $p_i-p_j$ is constant. A tail of the sequence is therefore one fixed polynomial plus a uniformly convergent sequence of constants, and its limit is a polynomial. The converse is immediate from the constant sequence. ThusThis is the globally uniform limit of real polynomials criterion.