I'm trying to prove that if $f\colon\mathbb{R}\to\mathbb{R}$ is continuous function and there is a sequence of polynomials $p_n$ that converges uniformly to $f$ on $\mathbb{R}$, then $f$ is a polynomial itself.
It smells like somewhat near the Weierstrass approximation with polynomials, but I have no idea how to use uniform convergence on the entire real line, since this statement is wrong on any bounded interval.
Any hint or solution will be appreciated.