Converge in $L^p$ implies convergence in $L^q$

Question

Let $(E,M,\mu)$ be a measured space. Let $p,q \in[1,\infty]$ with $p\neq q$, and let $(f_n)$ be a sequence of functions in $L^p\cap L^q$. Assume that $f_n\to 0 $ as $n \to \infty$ and $（f_n）$ is Cauchy in $L^q$. How can we show $f_n\to 0$ in $L^q$?

Answer 1

For $p<q$, convergence in $L_{q}$ implies convergence in $L_{p}$ (Hint: Prove this with Jensen's Inequality)