Given sequences (a_1, a_2, ...), (b_1, b_2, ...) of real numbers with a_i > 0 for all i, define h: \mathbb{R}^\omega \rightarrow \mathbb{R}^\omega by the equation h((x_1, x_2,...))=(a_1x_1+b_1, a_2x_2+b_2,...).
1. Show that if \mathbb{R}^\omega is given the product toopology, h is a homeomorphism.
2. What happens if \mathbb{R}^\omega is given the box topology?
3. If \mathbb{R}^\omega is given the uniform topology, under what conditions on the numbers a_i, b_i is h continuous? a homeomorphism?

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