If you’ve done mathematical research, you are probably aware of the following minefield (especially if you study non-compact spaces):
1. Establish for all . Note: I am abusing the sign here; I mean “equivalence class equality”.
2. Then try to conclude that only to find out….that the limit fails to exist, though it might exist if you put restrictive conditions on either the or on HOW the equivalence is obtained. 😦
This really shouldn’t surprise me at all; after all one of the things we teach our advanced calculus students is this example:
let . Index the rational numbers by
Let if , if . Then, while it is true that for all , the limit fails to be Riemann integrable (though it is Lebesgue integrable).
How quickly I sometimes forget the basics. 🙂