Real Numbers: Show Base b Analogy Properties

[Diffy]

Question:
Decimal (10-nary) expansions of real numbers were defined by special reference to the number 10. Show that the real numbers have b-nary expansions with analogous properties, where b is any integer greater than 1.

Attempt at solution:
I think if I show that there is a bijective function between the real numbers base ten, and any other base that will show they have analogous properties.

so let a₀.a₁a₂... be any real number, where a₀is any integer and a_i i >0 and i /in {0,1,2,...,9}.

Then it has been shown (in the book) that this can be represented as
a₀ + a₁/10 + a₂/10² + ...

I think now I need to show that this number can be changed into base b which I am not quite sure how to do. And even once I have done that, I am not sure that I am any closer to solving the problem.

Any help is appreciated.

Thanks

-Dif

 

[mathman]

The above proof for 10 carries over to base b with the following changes.
1) range of a's is (0,b-1).
2) denominators are powers of b rather than powers of 10.