Let f in C[a,b]. Show there exists a sequence of polynomials {p} such that (sup norm) ||p-f|| -> 0 for all n.
I think fixing g less than f and approximating that g with the polynomials is where to begin, but I don't see how to guarantee that the polynomials stay below f (or below g, for that matter). It looks like a Weierstrass Approximation Problem but I don't think it can be proved using that.
Hint: For each $n \in \mathbb{N}$. Consider the function \begin{align} f_n(x) = f(x) -\frac{1}{n} \end{align} then by Weierstrass approximation theorem there exists a polynomial, call it $P_n(x)$, such that \begin{align} \|P_n(x) - f_n(x)\|_\infty <\frac{1}{n} \end{align} which mean \begin{align} f(x)-P_n(x) = f(x)-\frac{1}{n}-P_n(x) + \frac{1}{n}>-\frac{1}{n}+\frac{1}{n}=0. \end{align} Let $n\rightarrow \infty$, then you will have your desired result.