TITLE: Uniform convergence of a sequence of polynomials
QUESTION [12 upvotes]: If a sequence of polynomials converge uniformly in $\mathbb{R}$ to $f$, is $f$ necessarily a polynomial?

REPLY [11 votes]: Claim: If $\{P_n\}$ is a sequence of polynomials which converges to $0$ uniformly on an unbounded subset $S$ of the real line, then $P_n$ is constant for $n$ large enough.

Indeed, $\sup_{x\in S}|P_n(x)|<1$ for $n$ large enough, hence $P_n$ is bounded on $S$. Fix such a $n$.  Let $x_k\in S$, $|x_k|\to +\infty$. If $P_n$ is of degree $d$, then $1>|P_n(x_k)|\geqslant |c_nx_k^d|$, where $2c_n$ is the leading term. It enforces $d=0$.
Now, we apply the preceding result to $P_{n+1}-P_n$.