= 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 $n\geq N$, 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.
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