Approximation in $L^p$ spaces

Question

Let $X$ denotes the span {$x^n:n \ge1 $}. Is it true that $X $ is dense in $L^1([0,1])?.$ I showed that $X$ is dense in the space of continuous functions that vanishes at zero. I also know space of continuous functions with compact support is dense in $L^p$ space. I guess it not true but my friend told me it is true. Anybody's help would be appreciated

Answer 1

Hint: given any continuous function on $[0,1]$, you can approximate it in the $L^1$ norm by continuous functions that vanish at $0$.