Consider the vector space $C([0,1])$ of real-valued continuous functions on $[0,1]$ endowed with the standard norm: $$ \Vert f\Vert_2 = \sqrt{\int_0^1 f(x)^2 dx}.$$ I know that this normed space is not complete.
Is this because the function $f_(x) = x^n$ converges to the discontinuous function which is zero on $[0,1)$ and $1$ at $x=1$?