# Thread: convergence in Lp space

1. ## convergence in Lp space

Let $(g_n)$ be a sequence that does not convergence in $L_p$ to a function $f\in L_p$ and let $(h_n)$ be a subsequence of $(g_n)$ that convergence in $L_p$ to $f$.
I know if it is the case where convergence is in $\mathbb{R}$,it is trivially true.But,what about in $L_p$?
2. Also (trivially) true, consider $\{g_1,f,g_2,f,\ldots\}$