Is it not correct that the sequence converges to the zero function on $[0,1]$ which is in $C[0,1]$ in the $L^1$ sense, that is: the area between the $n$th monomial and the $0$ function goes to $0$ as $n \to \infty$?
My prof gave it as a proof that the normed space $C[0,1]$ with $L^1$ norm is not complete, but I don't get why because there exists a limit in $C[0,1]$.