The value of a zero-sum two-person game with infinite number of stages can be defined either directly or as the limit of the values $v_n$ of the truncated games with $n$ stages. It is shown that these two concepts are not equivalent. There are games in which $\lim v_n$ exists but which do not have values as infinite stage games.
"On the Notion of Value for Games with Infinitely Many Stages." Ann. Statist. 1 (4) 791 - 796, July, 1973. https://doi.org/10.1214/aos/1176342477