Infinitely many prisoners

I heard this today at lunch: there are infinitely many prisoners $1,2,3,\dots$ each with a white or a black hat. Each prisoner can see everybody’s hat except his own. The deal is as folllow, the prisoners can agree on a common strategy before receiving the hat but cannot communicate afterward. If all but finitely many guess the color of their own hat correctly they all go free. The prisoners consider sequences of zeros and ones indexed by $1,2,3,\dots$, where $1$ stands for black and $0$ stands for white. They put an equivalence relation on such sequences where two sequences are equivalent if they differ in only finitely many places. They then use the Axiom of Choice and pick a representative sequence in each class. Everybody knows and agrees with the given picks. Now when the time comes for a prisoner to guess the color of his own hat, he looks out at the sequence ahead of him and determines its equivalence class. Then he guesses according to the color he would have if the current sequence was the actual representative agreed upon. In so doing only finitely many prisoners will guess wrong. So they all go free.