We know that the cosets $Hg$ form a partition of $G$ (see the coset entry for proof of this.) Since $G$ is finite, we know it can be completely decomposed into a finite number of cosets. Call this number $n$ We denote the $i$ coset by $Ha_i$ and write $G$ as

$$ G = Ha_1 \cup Ha_2 \cup \cdots \cup Ha_n $$

since each coset has $|H|$ elements, we have

$$ |G| = |H|\cdot n $$

and so $|H|$ divides $|G|$ which proves Lagrange's theorem. $\square$