From Theorem 3.13 on page 69 of Rudin's real and complex analysis, I learned that the set $S$ of all simple functions with finite support is dense in $L_p$ for $1 \leq p<\infty$. Obviously $S\subset L_q$ for all $q>0$. Then it seems obvious to conclude that $L_q \cap L_p$ is dense in $L_p$ simply because S is dense in $L_p$ and $S \subset L_q \cap L_p$.
I think the argument is quite simple here, but I couldn't find any resource to confirm this claim. Am I missing something in my argument?