=+3. Consider a supercritical branching process Xn with progeny generating function Q(s) and extinction probability s∞. Show that Pr(1 ≤ Xn ≤ k)sk

∞ ≤ Q n(s∞)

for all k ≥ 1 and that Q

n(s∞) = Q

(s∞)

n.

Use these results and the Borel-Cantelli lemma to prove that Pr 

limn→∞ Xn = ∞

= 1 − s∞.