If $\displaystyle <f_n>$ is a sequence in $\displaystyle L^p$ converging in $\displaystyle L^p$ to anfin $\displaystyle L^p$, that is, $\displaystyle \|f_n-f\|_p\to 0$ as $\displaystyle n\to\infty$, do we necessarily have $\displaystyle \lim\limits_{n\to\infty}\|f_n\|_p=\|f\|_p$? If not, could you please come up with a counterexample?

If $\displaystyle <f_n>$ is a sequence in $\displaystyle L^p$ satisfying $\displaystyle \lim\limits_{n\to\infty}\|f_n\|_p=\|f\|_p$ where $\displaystyle f\in L^p$, do we necessarily have $\displaystyle <f_n>$ converges in $\displaystyle L^p$ tof(as defined above)? If not, could you please come up with a counterexample?

Thanks!