Let $p<q$ be positive integers (with the allowance that $q$ may be $\infty$). How can we show that the sum of $L^p$ and $L^q$ is a Banach space under the norm $\|f\|=\inf\{\|g\|_p+\|h\|_q: g+h=f\}$?
Let $\{f_n\}$ be a sequence in $L^p+L^q$, such that $$\sum_{n=1}^\infty \|f_n\|<\infty.$$
We would like to conclude that this implies that $\sum_{n=1}^\infty f_n$ converges to something in $L^p+L^q$, at which point it follows that $L^p+L^q$ is complete by a basic theorem. We could easily do this if for each $f_i$ it were possible to express $f_i=g_i+h_i$, where $g_i\in L^p, h_i\in L^q$ and $\|f_i\|=\|g_i\|_p+\|h_i\|_q$. Though we know Cauchy sequences in $L^p$ converge, it is not clear that for a given $f$, all sequences (or some sequence) $\{g_n+h_n\}$ such that the $(\|g_n\|_p+\|h_n\|_q)\to \|f_n\|$ have the property that $\{g_n\},\{h_n\}$ converge in their respective spaces. It seems possible to imagine the sum converging without the summands converging in their respective spaces.
Any help would be much appreciated.