Cauchy sequences on $L^p$ and $L^q$

Question

Let $\{u_m\}_m$ be a Cauchy sequence on $L^p(\mathbb{R}^n)$ and $L^q(\mathbb{R}^n)$. Let suppose $u_m\longrightarrow u$ on $L^p(\mathbb{R}^n)$.

How do I prove that $u_m\longrightarrow u$ on $L^q(\mathbb{R}^n)$?

Answer 1

Hint: Observe $L^q(\mathbb{R}^n)\cap L^p(\mathbb{R}^n)$ is a Banach space, i.e. it's complete.

Edit: Okay. Since $\{u_m\}$ is a Cauchy sequence in $L^q\cap L^p$, which is complete, then there exists $w \in L^q\cap L^p$ such that $\|u_m-w\|_{L^q\cap L^p} \rightarrow 0$ as $m \rightarrow \infty$. Next, since $u_m\rightarrow u$ in $L^p$ then we have \begin{align} \|w-u\|_{L^p} \leq \|w-u_m\|_{L^p\cap L^q}+\|u_m-u\|_{L^p} \rightarrow 0 \end{align} which means $w=u \in L^p$. In particular, we have that \begin{align} \|u_m-u\|_{L^q} \leq \|u_m-u\|_{L^p\cap L^q} = \|u_m-w\|_{L^q\cap L^p}\rightarrow 0. \end{align}