Before the diagonalization argument, Cantor had another proof of the uncountability of the reals. Looking it up again, it's actually pretty much the Alice-Bob game!
(To see the correspondence, it's helpful to think of limits as being a kind of game. You claim the limit is a particular value, another party challenges you with an epsilon, and then you respond to the challenge by saying an index in the sequence after which all the entries are within epsilon of the claimed limit.)
See theorem 2: https://www.maa.org/sites/default/files/pdf/upload_library/2...
(To see the correspondence, it's helpful to think of limits as being a kind of game. You claim the limit is a particular value, another party challenges you with an epsilon, and then you respond to the challenge by saying an index in the sequence after which all the entries are within epsilon of the claimed limit.)