= Canonical-height proof of Mordell-Weil finite generation
For an <elliptic curve> over a <number field>, its nonnegative <canonical height of an elliptic curve> has finite bounded subsets by the <Northcott theorem> and its bounded difference from half the naive logarithmic height. By the <Weak Mordell-Weil theorem>, choose representatives $R_i$ modulo $2E(K)$ and put $H=\max_i\widehat h(R_i)$. If $P=2Q+R_i$, the <Cauchy-Schwarz inequality> for the height pairing gives $\widehat h(Q)\leq(\widehat h(P)+H)/2$. Iterating brings every point into the finite set of height at most $H+1$. This set and the $R_i$ generate $E(K)$, proving the <Mordell-Weil theorem> without assuming finite generation when constructing the height.
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