Suppose $f$ is a continuous function on $\mathbb R$. Show that we can approximate $f$ uniformly by a sequence of polynomials on any bounded subset of $\mathbb R$.
My attempt is as follows:
$f$ being continuous on $\mathbb R$ is finite valued and well-defined at all points in $\mathbb R$, in particular on any bounded subset. Now consider any sequence of compact intervals converging to that subset. Then for each such compact interval, consider a sequence of polynomials converging to $f$ uniformly.
I am stuck after this...is this even the correct approach? Can we use the concept of Power Series or Taylor Series here? If so, how?