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

Fix $P$ and put $e_m=h_x(mP)-m^2{h}_x(P)$ for $m\ge 0$. Then $e_0=e_1=0$. Apply the given approximate parallelogram identity with its two arguments $mP$ and $P$. The quadratic terms cancel exactly, leavingSet $d_m=e_m-e_{m-1}$. Starting from $d_1=0$, the inequality gives $|d_m|\le (m-1)C$. Summing from one to $m$ therefore yieldsThus $C_m=m(m-1)C/2$ works independently of $P$, including $m=1$.

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.