= Solution
Pass to the finite extension $L=K(E[n])$. Part c shows that the kernel of $E(K)/nE(K)\to E(L)/nE(L)$ is finite, so it is enough to control the image after $n$-torsion becomes rational. The <Kummer map of an elliptic curve>
$$
E(L)/nE(L)\hookrightarrow H^1(L,E[n])
$$
associates to $P$ the finite extension generated by one $n$-division point of $P$.
Let $S$ contain the places over $n$, all archimedean places, and all places of bad reduction. The local theory of elliptic curves shows that these Kummer classes are unramified outside $S$. Because $E[n]$ is finite and constant over $L$, such classes are controlled by finitely many $S$-unramified power classes. Finiteness of the class group and finite generation of the unit group make that power-class group finite. The Kummer image is therefore finite, proving the <Weak Mordell-Weil theorem> that $E(K)/nE(K)$ is finite; this is the <Kummer-theoretic proof of the weak Mordell-Weil theorem>.
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