Uniform convergence of a sequence of polynomials

Solution 1:

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 $P_{n+1}-P_n$.

Solution 2:

Yes. Any two functions $f,g$ with $\sup|f(x)-g(x)|<\infty$ must grow at the same rate. A degree $d$ polynomial $g$ grows at rate $O(|x|^d)$. So if polynomials $g_1,g_2,\dots$ converge uniformly to $f$ then all of $f,g_1,g_2,\dots$ grow at rate $O(|x|^d)$ for some integer $d$, and thus $g_1,g_2,\dots$ must have degree $d$. By Lagrange interpolation, for each $j$ the $j$'th coefficient of $g_k$ converges as $k\to\infty$ to some $c_j$, and $f$ is the polynomial with these limits $c_j$ as coefficients.