Infinite Coins Tossed Infinitely Often 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

The answer to the question in your first sentence is yes, with probability $1$. However, the assertion of your second sentence, beginning with "I.e.", is false, also with probability $1$. The same applies to the question and assertion of your second paragraph.
For any coin, the probability that a given finite sequence will occur infinitely often is $1$, while the probability that the finite sequence will occur sequentially infinitely often is $0$.
