Okay, how is this for a definition of convergence:
An infinite sequence xs :: [()] converges if there exists an n such that for every m > n, xs !! m = xs !! n. In other words, this is a bunch of programs which either halt or don’t, and after some point, either they all halt or they all don’t.
Then an infinite sequence xs :: [a] converges if map f xs converges for every f :: a -> ().