How to show that the sum of $L^p$ spaces is Banach.

Hint: for each $n$, choose $g_n \in L^p$, $h_n \in L^q$ such that $f_n = g_n + h_n$ and $||g_n||_p + ||h_n||_q \le ||f_n|| + 2^{-n}$.

You want in fact to show the following result:

Let $(X_1,||\cdot||_1)$ and $(X_2,||\cdot||_2)$ two Banach spaces such that $X_i\subset V$ where $V$ is a vector space. We define $X=\{x_1+x_2,x_1\in X_1,x_2\in X_2\}$ endowed with the norm $||x||_X:=\inf\{||x_1||_1+||x_2||_2,x=x_1+x_2\}$. Then $(X,||\cdot||_X)$ is a Banach space.

To see that, take $\{x^{(n)}\}$ a Cauchy sequence in $X$. We can extract a subsequence, denoted $\{y^{(k)}\}$ such that $||y^{(k+1)}-y^{(k)}||_X\leq 2^{-k}$ for all $k$. Let $(y_1^{(k)}, y_2^{(k)})\in X_1\times X_2$ such that $||y^{(k+1)}-y^{(k)}||_X+2^{-k}\geq ||y_1^{(k)}||_{X_1}+||y_2^{(k)}||_{X_2}$ and $y^{(k+1)}-y^{(k)}=y_1^{(k)}+y_2^{(k)}$. Since $X_1$ and $X_2$ are Banach spaces we can define $y_1:=\sum_{k=0}^{+\infty}y_1^{(k)}$ and $y_2:=\sum_{k=0}^{+\infty}y_2^{(k)}$. We have $$y^{(n+1)}=y^{(0)}+\sum_{k=0}^ny^{(k+1)}-y^{(k)}+y^{(0)}=\sum_{k=1}^ny_1^{(k)}+\sum_{k=1}^ny_2^{(k)}+y^{(0)},$$ which shows that $y^{(n)}$ converges to $\sum_{k=0}^{+\infty}y_1^{(k)}+\sum_{k=0}^{+\infty}y_2^{(k)}+y^{(0)}$.