We have an urn with two marbles numbered $1$ and $2.$ We pick a marble randomly, write down its number and return it to the urn. Then we add a marble with the number 3 to the urn, choose one of the three marbles randomly, record its number and return it to the urn. We repeat this over and over: before the $(k+ 1)$st pick we add a marble with with the number $k + 2$ in the urn (so that it contains the numbers $1, \dots , k + 2)$, choose a marble randomly, record its number (that's the $(k + 1)$st pick) and return it to the urn. At the end of the experiment we have an infinite sequence of integers. Show that with probability one the marble with the number $1$ will be picked at some point.
It was recommended to break up the event into disjoint pieces. But I have no idea how to do this. Also, does this mean that all drawings must give a different number, with say $1$ in the $n$th draw?
Any hints are much appreciated.