Does monomials approximate the constant function in the sense of $L^2$?

Question

I'm trying to show that $\{x,x^2,x^3,\dots\}$ approximate the constant function in the sense of $L^2[0,1]$-convergence; i.e. that there exists a sequence of polynomials $p_n$ with $p_n(0) = 0$ such that $$ \Vert 1 - p_n \Vert_2^2 = \int_0^1 \vert 1 - p_n(x) \vert^2 dx \to 0. $$ How do one go about proving such a statement (or disproving it)? Since the $L^2$ norm is not that easy to manipulate, I tried using the fact that $\Vert \cdot \Vert_2 \leq \Vert \cdot \Vert_\infty$, however this fails of course since $\Vert 1 - \sum c_i x^i \Vert_\infty$ is unbounded... Any hints on how to solve this?

Answer 1

Choose a continuous function $f$ so $f(0)=0$ and $||1-f||_2<\epsilon$. Choose a polynomial $q$ so $||f-q||_\infty<\epsilon$. Let $p=q-q(0)$; then $||p-q||_\infty<\epsilon$. So $||1-p||_2\le 3\epsilon$.