If an infinite number of coins are tossed infinitely often, is it true that there will be infinite subsets of those coins that repeat any finite sequence of heads/tails infinitely often? I.e., infinitely many coins will always produce heads, infinitely many always produce tails, infinitely many produce HTHTHT..., THTHTH..., HHTHHTHHT..., TTHTTHTTH..., etc.
And on each toss of all the coins, would some infinite subset of coins begin reproducing a finite sequence infinitely often? I.e, infinitely many coins that had previously produced irregular sequences of HT would begin producing HTHTHT..., THTHTH..., et