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Take the largest element a of S0. Then S1 will contain a+1, S2 has a+2 and so on. For each N in Z+, the set SN contains some element of S0, it's greatest, +N.
To show that SN also contains all of S0, we first assume S0 is a singleton. Then the next set contains S0 and it's successor. As new elements are added the singleton remains as a-1 is never there and new elements are added as old ones are dropped.
For each new element added to S0 at the beginning, as long as there is a gap of 1 between them, they all continue on into SN. If there is no gap between two elements, the lesser stays and the greater disappears in the next set. But then afterwards since only a-1 exists, a comes back again, oscillating on and off. If we have a long chain of nongapped numbers, only the smallest remains, and one greater than all comes in. Then the set rebuilds itself going: 1, 12, 1-3, 1234, 1---5, 12--5, 1-3-5, 123456, then continuing by returning to the sequence of odds 1-3-5-7-9... By adding and deleting the larger of the pairs fuck it's hard to explain
