Putnam 1968 A3

Let $S$ be a finite set and $P$ the set of all subsets of $S$ . Show that one can label the elements of $P$ as $A_i$ such that (1) $A_1 =\emptyset$ . (2) For each $n\geq1$ we either have $A_{n-1}\subset A_{n}$ and $|A_{n} \setminus A_{n-1}|=1$ or $A_{n}\subset A_{n-1}$ and $|A_{n-1} \setminus A_{n}|=1.$