Let be a topological space. let be the collection of all the ultrafilters on .
we say that a set is open and belongs to the base of if
there is an open , such that .
Let be the space of ultrafilters (filters on with the discrete topology) as explained above.
Prove that if a sequence converge in for every large enough .
any help is appreciated...
Another way to phrase this problem is to say that every convergent sequence in the Stone–Čech compactification of N is eventually constant. I believe that this is a standard result in the theory of the Stone–Čech compactification, but I do not know of a proof. You might try looking here for ideas and further references.