Given sequences of real numbers with for all , define by the equation .

1. Show that if is given the product toopology, is a homeomorphism.

2. What happens if is given the box topology?

3. If is given the uniform topology, under what conditions on the numbers is continuous? a homeomorphism?

For part 1, I know how to show is bijective, but having trouble showing and continuous. I don't know how to do 2 and 3. Can I get some help please?