Which Dominoes Remain Standing

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Problem 10: (Which Dominoes Remain Standing)

Suppose that Jon has set up an infinite number of dominoes, with the dominoes numbered $1,2,3, \ldots$. The dominoes are set up so that if the $k$th domino falls, then the $(k+1)$st domino will also fall. So if the 7th domino falls, then the 8th must fall as well. Jon knocks down the first domino, which starts causing other dominos to fall. Which dominos fall? Which dominoes remain standing? Make sure you prove your result. The well ordering principle will come in handy.

Suggestion: Use set builder notation to help you, so let $F=\{n\in \mathbb{N}\mid \text{domino $n$ fell}\}$ and $S=\{n\in \mathbb{N}\mid \text{domino $n$ remains standing}\}$. Then make some claims and prove they are correct.

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