Binary Decimals - Dual Expansion

[clg211]

Hi,

I'm trying to prove that there's a bijection between the open interval (0,1) and the set of all sequences whose elements are 0 or 1 in order to show cardinality continuum.

So let C={a1, a2, a3,...|ai is either 0 or 1} which is the set of all sequences of 0's and 1's
and let D={0.b1b2b3...|bi is either 0 or 1} which is the set of all binary decimals on the closed interval [0,1]
I think it's pretty clear that there's a bijection between these two sets.

Then the open interval (0,1)=D\{0.000..., binary decimals with tails of repeating 1's} which is the part that gives me problems.

I'm trying to get rid of the dual expansions by getting rid of the tails of 1's. For example, I have both 0.1 and 0.0111... in D which are the same number, and I want to get rid of 0.0111...

Can someone please explain to me the rational numbers that will have this dual expansion in binary? A denominator of what form will cause this? I think this will help me in explicitly defining (0,1) in terms of D. Any other thoughts on what I've already stated would be appreciated as well.

 

[CRGreathouse]

Repeating tails of 1 in the binary expansions correspond to numbers with denominators that are powers of two.