Globally uniform limit of real polynomials
= 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.