= Solution
Let $x:E\to\mathbb P^1$ be the $x$-coordinate map and define the naive logarithmic height by
$$
h(O)=0,
\qquad h(P)=\frac12\log H(x(P)).
$$
The duplication formula induces a degree-four morphism on the $x$-line, so part (a) gives
$$
h(2P)=4h(P)+O(1)
$$
uniformly in $P$. The telescoping sequence $4^{-n}h(2^nP)$ is therefore Cauchy. Define the <canonical height of an elliptic curve> by
$$
\widehat h(P)=\lim_{n\to\infty}4^{-n}h(2^nP).
$$
Summing the geometric error series proves that $|h(P)-\widehat h(P)|$ is bounded. If another function has this bounded-difference property and scales by four under doubling, evaluating the bounded difference at $2^nP$ and dividing by $4^n$ proves uniqueness.
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