Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 125 / 4 / c / ii / Solution

Replacing $[2^r]$ and $4^r$ by $[3^r]$ and $9^r$ gives the same canonical height. Indeed $h([3]P)=9h(P)+O_E(1)$, and the same telescoping argument constructs a quadratic height differing from the original by a bounded function. A bounded quadratic function is zero, so the two limits agree.