**Proof.** Let $X_i$ the random variable that is $1$ when point $p_i$ causes a change of $\delta$ and a recomputation of the data structures, and $0$ if not. It is easy to show that the cost is $O(n + \sum_{i=1}^{n} i X_i)$, since on the $i$-th step we are considering only the first $i$ points. However, turns out that $\Pr(X_i = 1) \le \frac{2}{i}$. This is because on the $i$-th step, $\delta$ is the distance of the closest pair in $\{p_1,\dots,p_i\}$, and $\Pr(X_i = 1)$ is the probability of $p_i$ belonging to the closest pair, which only happens in $2(i-1)$ pairs out of the $i(i-1)$ possible pairs (assuming all distances are different), so the probability is at most $\frac{2(i-1)}{i(i-1)} = \frac{2}{i}$, since we previously shuffled the points uniformly.

We can therefore see that the expected cost is