Here’s another proof, this time by induction:
Induction Hypothesis: (Not necessarily true yet) Among any card set of N cards, every move sequence terminates after a finite number of moves.
Using the Induction Hypothesis, prove the Induction Hypothesis for N+1:
Induction Step: The leftmost card of any set cannot be flipped back, once it has been flipped face-up, so it can be flipped at most once. Therefore, the maximal sequence in a set of N+1 cards cannot be longer than "maximal sequence among the N rightmost cards + flip the leftmost card + another maximal sequence among the N rightmost cards". Particularly, using the Induction Hypothesis, every move sequence among a set of N cards terminates, and therefore every move sequence among a set of N+1 cards also terminates.
Base case: Any set consisting of only 1 card allows for only terminating sequences. Proof: the possible sets are "face up" and "face down", which terminate after 0 and 1 moves, respectively.
Now, the Base Case proves that the Induction Hypothesis is true for N=1, and the Induction Step proves that if the Induction Hypothesis is true for some N, it is also true for N+1.
Conclusion: Therefore, by induction, for all integers N >= 1, the Induction Hypothesis is true.