A sequence of polynomials converging pointwise to a given continuous function on $\mathbb R$ convergence being uniform in every compact interval.

Question

How to prove that given any continuous function from $\mathbb R$ to $\mathbb R$ we can get a sequence of polynomials converging to that function pointwise and the convergence is uniform on any compact set in $\mathbb R$. The problem is taken from

Answer 1

From Weierstrass theorem, $\forall n \in \Bbb{N}$ exists $p_n$ polynomial such that $$\sup_{x \in [-n,n]}|p_n(x)-f(x)| \leq \frac{1}{n}$$

Clearly for every bounded interval $[a,b]$,for all $n>\max\{|a|,|b|\}$ we have that $$\sup_{x \in [a,b]}|p_n(x)-f(x)|\leq \sup_{x \in [-n,n]}|p_n(x)-f(x)| \leq \frac{1}{n} \to 0$$

Answer 2

For each $n\in \Bbb N$ consider a polynomial $p_{n}$ with $$\sup_{x\in [-n,n]}|f(x)-p_n(x)|<\frac{1}{n}.$$ Take the sequence $\{p_n\}.$