Density of $C_c^\infty(\Omega)$ in $L^p(\Omega)$ for a bounded open set $\Omega$ - any detailed proof?

Question

Let $\Omega$ be an open, bounded domain in $\mathbb{R}^n$ (NOT necesasrily with smooth boundary).

Now, let $C_c^\infty(\Omega)$ be the space of smooth functions on $\Omega$ with compact support.

Then

How to prove that $C_c^\infty(\Omega)$ dense in $L^p(\Omega)$ for each $p \in [1,\infty)$?

This post has almost the same question and answer, but no details of proof.

Could anyone please provide some details (or at least sketch of) how to do the approximation correctly?