A math checkerboard question

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Suppose you have an infinite row of squares and two checkers. The checkers take turns responding to coin tosses; heads a checker moves right and tails a checker moves left.
Wherever the checkers begin, eventually, given an infinite time, they will meet. Obviously. After all there is infinite time.
Now suppose you have, instead of a row of squares, an infinite checkerboard and a 1-8 spinner.
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Checker movements are determined by the spinner: 1 means move straight ahead; 2 means move diagonal ahead right, etc.

Wherever the checkers begin, eventually, given an infinite time, they will meet. Obviously. After all her is an infinite time.

Now the point. Suppose we have an infinite number of infinite checkerboards on top of one another and an appropriate device for determining checker movement.

Wherever the checkers begin, eventually, given an infinite time, they will meet. Obviously. After all there is an infinite time

Well, No. I have read that, on an infinity of infinite checkerboards on top of one another, the checkers will never meet, even in infinite time.

I don’t understand this. How can it be? There is an infinite time and yet the checker never meet. How can this be?