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The weak lower semicontinuity of the functional $$F\left(u\right)={\int }_{\Omega }f(x,u,\nabla u)\mathrm{d}x$$ 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 $W^{1,p}\left(\Omega \right)$. However, the known proofs use advanced instruments of real and functional analysis. Our aim here is to present a proof understandable even for students familiar only with the elementary measure theory.