MAYBE Lemma: In TMA if on day i a girl says “maybe” to boy b, she is guaranteed to marry a husband that she likes at least as much as b
(*) For all k 0, on day i+k the girl will say “maybe” to a boy she likes as much as b.
BASE: k=0 (true by assumption)
Assume (*) is true for k-1. Thus she has a boy as good as b on day i+k-1. The next day she will either keep him or reject him for some better. Thus (*) is true for k.