show that any continuous function can be approximated uniformly

Question

I do not know where to start because i have not dealt with a question like this before. I feel that i have to use the Stone-Weierstrass theorem, but im not sure how to use it.

Answer 1

Consider the function $g(x) = f(x)/x$. By Stone-Weierstrass you know that there exists a sequence of polynomials $p_k(x)$ uniformly approximating $g(x)$ on $[\epsilon,1]$. To prove that the sequence $q_k(x) = x\cdot p_k(x)$ uniformly approximates $f$ just notice that $$ |q_k(x) - f(x)| = |x\cdot p_k(x) - f(x)| = |x|\cdot|p_k(x) - g(x)| \le |p_k(x)-g(x)| \to 0 $$