Let $\displaystyle (g_n)$ be a sequence that does not convergence in $\displaystyle L_p$ to a function $\displaystyle f\in L_p$ and let $\displaystyle (h_n)$ be a subsequence of $\displaystyle (g_n)$ that convergence in $\displaystyle L_p$ to $\displaystyle f$.

Does any contradiction exists?

I know if it is the case where convergence is in $\displaystyle \mathbb{R}$,it is trivially true.But,what about in $\displaystyle L_p$?