If I know that converges uniformly on and converges pointwise on and , how can I show that converges uniformly on ?
In fact we can strongly generalize this. If converge pointwise on and uniformly on where then converges uniformly on .
A more interesting question is whether we can have a similar result for infinitely many points removable as well. I am not sure yet.
EDIT: There is a simple generalization for infinitely many point but it is not an interesting generalization: if for any the set of 's for each pointwise point is bounded then the sequence is actually uniformly convergent everywhere.