I want to show that a sequence of functions in $L^p$ converges $f_n\to f \in L^p$ if every subsequence has a further subsequence that converges to $f\in L^p$.
We are given that every subsequence has a further subsequence that converges to $f\in L^p$. My attempt is to suppose $f_n$ does not converge to $f$ in $L^p$. This means there is some $\varepsilon,\delta,$ and $\{f_{n_k}\}$ such that $$m(x\vert \|f_{n_k}-f\|_p>\delta)>\varepsilon$$ for all $k$. But we know that there is some $g_k\subseteq f_{n_k}$ such that $g_k\to f$ in $L^p$ since this is given, but this is a contradiction to the line above. Is this correct?
Yes, that's correct. Actually what you propose works for any topological space with the same reasoning.
Moreover, you can see this question for more information on nets.