Answer:
Look at the cards as numbers
Face Down = 1 and Face Up = 0

Initially there will be a sequence of 1’s as all the cards will be face down. After a while it would look something like this:
1 0 0 1 1 0 1 0
As we can see this is a Binary number, and a move that consists of turning a face down card face up and the card immediately to the right could be that a one followed by a one will turn into a zero followed by a zero; which would look like this:
1 0 0 0 0 0 1 0
Or it could be a one followed by a zero turning into a zero followed by a one
1 0 0 0 0 0 0 1
In either case we can see that the number in Binary is strictly decreasing, which means the sequence must terminate as you can’t keep taking away from a positive integer without it turning negative.