Coming down from infinity of the Kingman coalescent
= Coming down from infinity of the Kingman coalescent
For $t>0$, the number $B_t$ of blocks is finite almost surely. On a finite restriction, the expected time to reach $m$ blocks is at most $\sum_{j>m}2/[j(j-1)]=2/m$. <Markov inequality>, followed by increasing the restriction, gives $\mathbb P(B_t>m)\leq2/(mt)$. Let $m\to\infty$, then use rational times and monotonicity to obtain the assertion simultaneously for every $t>0$.