$L^1$-convergence implies $L^p$-convergence

Question

If $\{f_n\}$ is a sequence in $L^1$ and in $L^p$ where $p>1$, also $\{f_n\}$ converges to $f$ in $L^1$, does that imply $\{f_n\}$ converges to $f$ in $L^p$ as well?

Thank you.

Answer 1

No. For example, $\sqrt{n}\,\chi_{[0,1/n]}$ converges to $0$ in $L^1$, but not in $L^2$.