  The weak lower semicontinuity of the functional
  $$ F(u)=\int_{\Omega}f(x,u,\nabla u)\, dx$$
  is a classical topic that was studied thoroughly. It was shown that if the
function $f$ is continuous and convex in the last variable, the functional is
sequentially weakly lower semicontinuous on $W^{1,p}(\Omega)$. However, the
known proofs use 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.
