I'm trying to solve the following problem and I have this solution. So the solution shows that $f_n-f$ is not even in $L^1$ space, when we need $\|f_n-f\|_1\to 0$ to show that the sequence converges to $f$. However, from my knowledge this only shows that $f_n$ does not converge to $f$. But we need to show that $f_n$ does not have a limit in $(Y, \|\cdot\|_1)$. Or, is there a fact that if a sequence converges to some limit in the $L^1$ space, then it must be pointwise convergent to that limit?
