Proof of fundamental lemma of calculus of variation.

Suppose $\Omega$ is an open subset of $\mathbb{R}^n$ and let $L^1_\text{Loc}(\Omega)$ denote all locally integrable functions on $\Omega$ and $C^{\infty}_0(\Omega)$ for smooth functions whose support lie in $\Omega$. My teacher tells me the following statement:

Suppose $f\in L_{Loc}^1(\Omega)$ and$$\int_\Omega f\varphi=0,\quad\forall\varphi\in C^\infty_0(\Omega)$$Then $f=0\text{ a.e.}$ on $\Omega$.

It is known as fundamental lemma of calculus of variation. My teacher told me it suffices to prove this statement holds for the case $f$ is continuous. But I find it's not easy to deduce the lemma from the case $f$ is continuous. Could someone tell me how to do this or how to prove the lemma directly? Thanks a lot!