 Uniform limits of polynomials on the real line

If real polynomials $p_n$ converge by uniform convergence to $f$ on all of the real line, then for some $N$ and constants $c_n$, $p_n=p_N+c_n$ for every $n\ge N$, with $c_n$ convergent. Indeed, a common uniform error below one gives $|p_n-p_N|<2$ everywhere. A bounded polynomial on the real line is constant, since every nonconstant polynomial grows in absolute value at infinity. Evaluating at zero identifies the limit of a sequence of constants, so $f=p_N+c$ is a polynomial. This conclusion depends on the entire real line being the domain.