SCHWEITZER 1982 Problem 10

Let $p_0,p_1,\ldots$ be a probability distribution on the set of nonnegative integers. Select a number according to this distribution and repeat the selection independently until either a zero or an already selected number is obtained. Write the selected numbers in a row in order of selection without the last one. Below this line, write the numbers again in increasing order. Let $A_i$ denote the event that the number $i$ has been selected and that it is in the same place in both lines. Prove that the events $A_i \;(i=1,2,\ldots)$ are mutually independent, and $P(A_i)=p_i$ .

T. F. Mori