I have 2 problems that are due tomorrow that I can't figure out. I couldn't find any example of such problems anywhere in my book or the internet.

Anyway here they are:

1. Suppose that a and b are disctinct real numbers such that a<b.

a) Prove that the set {x e : a<x<b} is uncountable.

b) Prove that the set {x e : a<x<b} is countably infinite

2. Use the fact that every real number has a unique decimal expansion that does not end in all 9's to prove that the interval (0,1) is an uncountable set.

Any help on these would be greatly appreciated.

Thanks.