But there's no previous state that can result in one person (I'll name them X) having all the money and everyone else having no money. There are 2 cases:
1) X had some money on the previous turn, giving a dollar to Y. Then Y has at least 1 dollar.
2) X had no money on the previous turn. Then by pigeonhole principle there exists a person Y who has at least 2 dollars. Then Y has at least 1 dollar on the current turn.
1) X had some money on the previous turn, giving a dollar to Y. Then Y has at least 1 dollar.
2) X had no money on the previous turn. Then by pigeonhole principle there exists a person Y who has at least 2 dollars. Then Y has at least 1 dollar on the current turn.