Test functions are dense in $L^p$?

Question

I was wondering about the following: If we say that the test functions are dense in $L^p$, does this imply that there is also always a sequence of them converging pointwise and in $L^p$ norm to such a $f \in L^p$?- I guess this is true and follows from the fact that $L^p$ implies that there is a subsequence converging pointwise a.e., am I right or is there anything I am missing?