Mordell–Weil theorem

The proof combines two ingredients. Weak Mordell–Weil: one shows $E(\mathbb{Q})/2E(\mathbb{Q})$ is a finite group, by embedding it (via Galois cohomology, using the $2$-descent map $P \mapsto (x(P)-e_1, x(P)-e_2, x(P)-e_3)$ for the roots $e_i$ of the cubic) into a group built from the class group and unit group of a related number field, both of which are known to be finite. Height descent: one attaches to each point a canonical height $\hat h(P) \ge 0$, a real-valued measure of arithmetic complexity satisfying $\hat h(2P) = 4\hat h(P)$ and for which only finitely many points have height below any given bound. Combining a finite set of coset representatives for $E(\mathbb{Q})/2E(\mathbb{Q})$ with the fact that repeatedly halving the height of any point (via the parallelogram law for heights) eventually lands in a bounded-height region shows every point is a $\mathbb{Z}$-combination of the finitely many representatives and finitely many bounded-height points — hence $E(\mathbb{Q})$ is finitely generated.