Let $(X, \Sigma, \mu)$ be a measure space, $p,q\in [1, + \infty ]$ and $(f_n)_{n\geq0}$ a sequence in $ L^p \cap L^q $. Suppose that $f_n$ converges to $0$ in $ L^p $ and that it's a Cauchy sequence in $L^q$.
Show that $f_n$ converges to $0$ in $L^q$.
I can already use that $L^p$ spaces are Banach spaces, but I don't really have any other clue, so any hint would be appreciated! Thank you!
I use the general fact that if $f_n \to f$ in $L^r$, then some subsequence $f_{n_k}$ converges pointwise a.e. to $f$.
Since $L^q$ is a Banach space, $(f_n)_n$ converges to some $g \in L^q$. Then some subsequence $f_{n_k}$ converges pointwise a.e. to $g$. Note that $f_{n_k}$ must converge to $0$ in $L^p$ since $f_n$ does. Hence, there is a further subsequence $f_{n_{k_m}}$ that converges pointwise a.e. to $0$. Of course, $f_{n_{k_m}}$ will converge pointwise a.e. to $g$ as well. Hence, $g = 0$ a.e., as desired.