Is $L^p \cap L^q$ dense in $L^r$?

Let $f\in L^r$, and define $f_n:=f\chi_{\{n^{-1}\leqslant |f|\leqslant n\}}.$ We have $f_n\in L^p\cap L^q$ for each $n$, and by monotone convergence, $\lVert f-f_n\rVert_r\to 0$ as $n$ goes to infinity.