  Consider a $\Lambda$-coalescent that comes down from infinity (meaning that
it starts from a configuration containing infinitely many blocks at time 0, yet
it has a finite number $N_t$ of blocks at any positive time $t>0$). We exhibit
a deterministic function $v:(0,\infty)\to(0,\infty)$ such that $N_t/v(t)\to1$,
almost surely, and in $L^p$ for any $p\geq1$, as $t\to0$. Our approach relies
on a novel martingale technique.
