Proving the existence of a sequence of polynomials convergent to a continuous function $f$.

Question

I need to show that if $f$ is continuous function ($f:\mathbb{R}\rightarrow \mathbb{R}$), then there exists a sequence of polynomials which converges to $f$ on any compact subset of $\mathbb{R}$.

I would appreciate any help/direction ...

Answer 1

Let $\epsilon_n = 1/n$.

Let $A_n$ be the closed interval $[-n,n]$.

Let $S$ be any compact set.

For each $\epsilon_n$, the Weierstrass Approximation Theorem states that there exists a polynomial $P_n(x)$ such that for any $x \in A_n$, $$|f(x) - P_n(x)| < \epsilon_n$$

The fact that $S$ is compact allows us to apply the Heine-Borel Theorem, which implies that $S$ is closed and bounded. The fact that $S$ is bounded means that for sufficiently large $n$, $S \subset A_n$.

You should now be able to prove that this sequence of polynomials $P_n$ converges to $f$ on $S$, using the fact that $$\lim_{n\to\infty} \epsilon_n = 0$$