Globally uniform limit of real polynomials

A function $f:{\mathbb{R}}^n\to \mathbb{R}$ is a uniform limit on all of ${\mathbb{R}}^n$ of real polynomials exactly when it is itself a polynomial. If $(p_i)$ converges uniformly, then $p_i-p_j$ is bounded on ${\mathbb{R}}^n$ for all sufficiently large $i,j$. A bounded real polynomial is constant, so a tail of the sequence differs from one fixed polynomial only by convergent constants.