 Kummer-theoretic proof of the weak Mordell-Weil theorem

The Kummer map of an elliptic curve embeds $E(K)/nE(K)$ into $H^1(K,E[n])$. Outside a finite set containing the primes over $n$ and the primes of bad reduction, the resulting classes are unramified. Class-group and unit finiteness implies that only finitely many such classes exist, so $E(K)/nE(K)$ is finite.