 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 $\mathbb{P}(B_t>m)\le 2/(mt)$. Let $m\to \infty$, then use rational times and monotonicity to obtain the assertion simultaneously for every $t>0$.