Past exam of the analytic number theory and elliptic curves course of the University of Oxford / 2008 / B / 8 / a / ii / Solution

The preceding bound at $m=2$ gives $|h_x(2Q)-4h_x(Q)|\le C$ for every point $Q$. For $a_n=4^{-n}{h}_x(2^{n}P)$, take $Q=2^{n}P$ to obtainFor $m>n$, summing the geometric tail givesThus the sequence is Cauchy and converges by completeness of the real numbers. The estimate also gives a quantitative bound on its distance from the limit.