The limit of a pointwise converging sequence of polynomial is smooth

Question

Let $(P_n)_{n\geqslant 0}$ being a sequence of real polynomials with non negative coefficients, that converge pointwise on $\mathbb{R}$ to a functoin $f$. Then:

$f\in \mathcal{C}^{\infty}$.

How one can prove that?

Answer 1

Your tags don't include complex analysis, but that approach gives a quick proof, so I'll post it.

We can consider each $P_n(x)=\sum_{m=0}^{M_n}a_{mn}x^m$ as defined on all of $\mathbb C,$ simply by replacing $x\in \mathbb R$ with $z\in \mathbb C.$ Suppose $|z|\le R.$ Then

$$|P_n(z)| = |\sum_{m=0}^{M_n}a_{mn}z^m| \le \sum_{m=0}^{M_n}| a_{mn}z^m|$$ $$\le \sum_{m=0}^{M_n}a_{mn}R^m = P_n(R).$$

But the sequence $P_n(R)$ converges, hence is bounded. This shows $|P_n(z)|$ is uniformly bounded on each disc $\{|z|\le R\}.$

By Montel's theorem, there is a subsequence $P_{n_k}$ that converges uniformly on each $\{|z|\le R\}$ to an entire function $g.$ Since $P_n \to f$ pointwise on $\mathbb R,$ the same is true of $P_{n_k}.$ It follows that $f=g$ on $\mathbb R.$ Thus $f$ is not just $C^\infty,$ but is the restriction of an entire function to the real axis.