Countable Infinite Prisoners Hat Guessing via Axiom of Choice

Scenario

A countably infinite number of prisoners, indexed $1, 2, 3, \dots$, are lined up in single file facing forward. Each prisoner wears a hat colored either Black ($0$) or White ($1$). Each prisoner $k$ can see all the hats worn by prisoners in front of them ($k+1, k+2, k+3, \dots$), but cannot see their own hat or any hat behind them ($1, \dots, k-1$). No communication or sound is permitted once hats are placed. All prisoners must simultaneously guess the color of their own hat.

Your Task

1. Define the tail equivalence relation $\sim$ on the Cantor space of binary sequences $\mathcal{S} = \{0, 1\}^\mathbb{N}$.
2. Invoke the Axiom of Choice to construct a collective strategy.
3. Prove that this strategy guarantees that at most a finite number of prisoners guess incorrectly, meaning an infinite co-finite tail of prisoners guess correctly.

Constraints

• Hats are chosen arbitrarily from $\{0, 1\}^\mathbb{N}$.
• Guesses are strictly simultaneous with no communication.

Examples If the sequence is $1, 0, 1, 0, 0, 0, \dots$ and the chosen representative is identical except at positions 1 and 3, exactly 2 prisoners guess wrong, and infinitely many prisoners guess correctly.