It is known that continuous functions are dense in an $L^p(X,\mu)$ space, with $p<1$, if the space $X$ is a locally compact Hausdorf space and $\mu$ is a regular Borel measure.
My question is if $f\in L^p$ and bounded, are there sequences of continuous functions $\{s_n\}_n$ and $\{i_n\}_n$, each of them converging to $f$ such that $s_n(x)\geq f(x)$ and $i_n(x)\leq f(x)$ for all $x\in X$?