Consider the well known Euler sequence of functions $x^n$ ($n=1,2,3\ldots$) on $[0,1]$. It is clear that it converges against $\chi\_{1}$, the characteristic function of the singleton $1$, in the $L^1$-Norm, and that this function does not lie in $C^0[0,1]$ is supposed to show that $C^0[0,1]$ is not complete. However it converges also against $0\in C^0[0,1]$. So is the argument really valid? Problem Fréchet?

EDIT: The main question is: does $x^n\rightarrow\chi\_{1}$ in $L^1$ refute that $C^0[0,1]$ is complete? After all we also have $x^n\rightarrow 0\in C^0[0,1]$ in $L^1$?