$L^p + L^r$ is complete

Question

From Folland, "Real Analysis, Modern Techniques and Their Applications", Section 6.1 exercise 4.

Let $1 \le p < r \le \infty$. Show that $L^p + L^r$ with norm $\|f\| = \inf\{\|g\|_p + \|h\|_r \mid f = g + h\}$ is a Banach space.

Showing that $\|\cdot\|$ is a norm is trivial. I don't know how to show completeness.

For a Cauchy sequence $\{f_n\}$ I wanted to consider $g_n \in L^p$ and $h_n \in L^r$ s.t. $f_n = g_n + h_n$ and $\|g_n\|_p + \|h_n\|_r \le \|f_n\| + \epsilon$. But the decomposition is not unique: even if $f_n$ and $f_m$ are close, it's not necessarily the case for $g_n, g_m$ and $h_n, h_m$.

Answer 1

Let $\{f_n\}$ be such that $\sum\|f_n\|<\infty$, and choose a decomposition $f_n=g_n+h_n$ such that $\|g_n\|_p+\|h_r\|_r\leq \|f_n\|+2^{-n}$. Then, $\sum\|g_n\|_p<\infty$ and $\sum \|h_n\|_r<\infty$. Since $L^p$ and $L^r$ are Banach spaces, what can you say (Folland surely has a theorem about this)? Therefore, what can you conclude about $f_n$?