We had the theorem about $C_c^{\infty}$ being dense in $L^p$, which, as I understand, means that if we already have an $L^p$ function, there is a $C_c^{\infty}$ function arbitrary close to it with respect to the $L^p$ norm.
But what if we have a candidate for being an $L^p$ function, and we can show that there is a $C_c^{\infty}$ function arbitrary $L^p$-close to the candidate?
It seems like we need a different statement since $A$ being dense in $B$ means that $$A\subset B \subset \bar{A}$$ where $\bar{A}$ can be larger than B.
Is the required statement true? How do we prove it?
If $(g_n)$ is a sequence in $C_c^{\infty}$ and $\|g_n-f\|_{L^p}\rightarrow0$, then $(g_n)$ is a Cauchy sequence and the limit $g\in L^p$ exists. It can easily be shown that $f=g$ almost everywhere.