= Solution
A <height function> measures arithmetic size, which is what makes an otherwise infinite descent terminate. For a <number field> $K$, normalize absolute values to extend the standard real and $p$-adic ones, and write $n_v=[K_v:\mathbb Q_v]$. The logarithmic <projective height> is
$$
h([a_0:\cdots:a_s])=\frac1{[K:\mathbb Q]}\sum_v n_v\log\max_j|a_j|_v.
$$
The <product formula> makes it independent of the chosen homogeneous coordinates, and the local-degree normalization makes it independent of the <number field> containing them. Over $\mathbb Q$, $h([a:b])=\log\max(|a|,|b|)$ for <coprime> integer coordinates. Set $h_x(P)=h(x(P))$ on an <elliptic curve>, with $h_x(O)=0$.
Two features are essential. First, the <Northcott theorem> says that points of bounded <projective height> and bounded field degree form a finite set. For points on a fixed <elliptic curve> over $K$, each $x$-coordinate has at most two preimages, so bounded $h_x$ gives finitely many points. Second, a degree-$d$ morphism of the <projective line> satisfies $h(f(z))=dh(z)+O(1)$, uniformly in $z$. The upper bound comes from evaluating its homogeneous <polynomials>; for the lower bound, their lack of a common zero gives a resultant identity bounding the input coordinates by the output coordinates at each place. Summing the local bounds gives the asserted uniform constant.
The duplication map on the $x$-line has degree four. Nonsingularity ensures that its numerator and denominator have no common projective zero. Therefore
$$
h_x(2P)=4h_x(P)+O(1).
$$
Telescoping defines the <canonical height of an elliptic curve>
$$
\widehat h(P)=\frac12\lim_{j\to\infty}4^{-j}h_x(2^jP),\qquad
\widehat h(P)=\tfrac12h_x(P)+O(1).
$$
The error in successive terms is bounded by a geometric series, proving convergence and the uniform bounded difference. It also gives $\widehat h(2P)=4\widehat h(P)$ and nonnegativity. The usual addition formula gives the approximate <height parallelogram identity> for $h_x$; equivalently, the unordered pair of sum and difference on the $x$-line has bidegree $(2,2)$. Applying that identity to $2^jP,2^jQ$ and passing to the limit gives
$$
\widehat h(P+Q)+\widehat h(P-Q)=2\widehat h(P)+2\widehat h(Q).
$$
In particular $\widehat h(nP)=n^2\widehat h(P)$. Its polarization is the <canonical height pairing>, a positive semidefinite bilinear form even before finite generation has been proved. The <Cauchy-Schwarz inequality> for this pairing gives
$$
\widehat h(P-Q)\leq2\widehat h(P)+2\widehat h(Q).
$$
Also $\widehat h(P)=0$ precisely for <torsion points of an elliptic curve>: one direction follows from periodic multiples, and in the other direction all multiples have bounded $h_x$, so the <Northcott theorem> makes two multiples equal. Bounded <canonical height of an elliptic curve> likewise gives a finite set of $K$-<rational points>.
Now the <Weak Mordell-Weil theorem> gives finitely many representatives $R_1,\ldots,R_s$ for $E(K)/2E(K)$. Put $H=\max_i\widehat h(R_i)$. Write any point as $P=2Q+R_i$. Then
$$
\widehat h(Q)=\tfrac14\widehat h(P-R_i)
\leq\tfrac12\widehat h(P)+\tfrac12H.
$$
Repeatedly applying this <height descent lemma> eventually reaches height at most $H+1$: after $j$ steps the height is at most $H+2^{-j}(\widehat h(P)-H)$. The set of points with height at most $H+1$ is finite. Reading the relations $P=2Q+R_i$ backwards shows that this finite set together with the $R_i$ generates $E(K)$. Consequently
$$
\boxed{E(K)\cong E(K)_{\mathrm{tors}}\oplus\mathbb Z^r,\qquad r<\infty.}
$$
\b[Heights turn weak Mordell-Weil finiteness into the Mordell-Weil theorem.] The <canonical height pairing> subsequently equips the free part with a positive definite <quadratic form>, useful for bounding searches and measuring independent generators; this interpretation is a consequence of the proof, not an assumption used in the descent.
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