The weak lower semi-continuity of the functional
$$
F(u)=\int_{\Omega}f(x,u,\nabla u)\dee x
$$
is a classical topic thath was studied thoroughly. It was shown that if the function $f$ is continuous and convex in the last variable, the functional is weakly lower-semicontinuous on $W^{1,p}(\Omega)$. However, most of the proofs used advanced instruments of real and functional analysis. Our aim here is to present proof that can be easily understood by students familiar only with the elementary measure theory.