= Solution
The natural map
$$
E(K)/nE(K)\longrightarrow E(L)/nE(L)
$$
has finite image by hypothesis. It remains to bound its kernel. If $P\in E(K)$ becomes $nQ$ for $Q\in E(L)$, then
$$
c_\sigma=\sigma Q-Q\in E(L)[n]
$$
is a one-cocycle for $G=\operatorname{Gal}(L/K)$. Changing $Q$ by an $n$-torsion point changes this cocycle by a coboundary, producing a well-defined map from the kernel to
$$
H^1(G,E(L)[n]).
$$
If its cohomology class is zero, subtracting the corresponding torsion point from $Q$ makes $Q$ Galois fixed, so $P\in nE(K)$. The map is therefore injective. Both $G$ and $E(L)[n]$ are finite, so this <group cohomology> set is finite. A finite kernel and finite image give
$$
\boxed{|E(K)/nE(K)|<\infty.}
$$
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