A population problem

Really a variation on a well-known theme, inspired by this and also recently appearing here.

Suppose there is an initially finite population consisting of males and females which breeds without multiple births according to the following rules. Each femaile continues to produce offspring until a male child is borne, and then ceases to breed. Show that (even in this ideal universe) with probability 1, a point in time will be reached after which no further reproduction is possible. We assume that births are iid with equal probability of a male or a female being born.

I suggest this as an exercise, and include a solution below the fold.