= Solution
The preceding bound at $m=2$ gives $|h_x(2Q)-4h_x(Q)|\le C$ for every point $Q$. For $a_n=4^{-n}h_x(2^nP)$, take $Q=2^nP$ to obtain
$$
|a_{n+1}-a_n|\le C4^{-n-1}.
$$
For $m>n$, summing the geometric tail gives
$$
\boxed{|a_m-a_n|\le\sum_{j=n}^\infty C4^{-j-1}=\frac C{3\cdot4^n}\longrightarrow0.}
$$
Thus the sequence is <Cauchy sequence>[Cauchy] and converges by <completeness of the real numbers>. The estimate also gives a quantitative bound on its distance from the limit.
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