C[0,1] is not complete with respect to integral norm [closed]
Solution 1:
$\int_0^{1} f_n(x)dx=\frac 1 n\int_0^{n} \frac 1 {y+1} dy=\frac 1 n \ln (n+1) \to 0$. Hence, this sequence converges in $L^{1}$ to the continuous function $0$. It is not a valid counter-example.