[Math] Is it true that any continuous function $f$ on $[0,\infty)$ can be approximated by polynomials

Question

I think it's true, but how to prove it? By weierstrass approximation theorem we can approximate uniformly any continuous function on a closed interval $[a,b]$ by a sequence of polynomial function i.e., given any $\epsilon>0$ there exists an $N\in \mathbb{N}$ such that $n>N$ $\implies$ $|p_n(x)-f(x)|<\epsilon$ $,\forall x\in [a,b].$ I am asking whether it is possible to approximate a continuous function $f$ by a sequence of polynomial functions on $[0,\infty).$

Thanks!

Answer 1

No, it is not true. Polynomials of degree $\ge1$ are unbounded on $[0,\infty)$. This implies that a bounded function cannot be approximated uniformly by polynomials on $[0,\infty)$.

Edit

This is added after reading Henning Makholm's comment.

If $\{p_n\}$ is a sequence of polynomials converging uniformly to a function $f$ on $[0,\infty)$, then $f$ is a polynomial.

Proof. By Cauchy's uniform convergence criterion, there exists $N\in\mathbb N$ such that $$ n\ge N\implies|p_N(x)-p_n(x)|\le1\quad\forall x\in[0,\infty). $$ $p_N-p_n$ is a bounded polynomialon $[0,\infty)$, thus it is a constant $c_n$: $$ p_n=p_N+c_n\quad n\ge N. $$ Since $\{p_n(x)\}$ converges for all $x\ge0$, $c_n$ also converges. Let $c$ be its limit. Then $f=p_N+c$.