I did something similar in a proof showing that monotone convergence implies least upper bound property.
Let the two distinct real numbers be a and b with a> b. Then a- b> 0 so [tex]\frac{1}{a-b}[/tex] exists and is positive. By the Archimedian property of integers (given any real number x, there exist an integer m with m>x) there exist an integer n with [tex]n> \frac{1}{a-b}[/tex].
But it is easy to prove by induction that, for all positive integers n, 2^{n}> n (Hint: first prove by induction that 2n>= n+1) so we have [tex]2^n> \frac{1}{a-b}[/tex]. That means that 2^{n}(a- b)= 2^{n}a- 2^{n}b> 1.
Since the distance between those two numbers is greater than 1, there exist an integer, k, such that 2^{n}b< k< 2^{n}a. Dividing through by 2^{n} gives [tex]b< \frac{k}{2^n}< a[/tex].
I was thinking more along the lines of:
WOLOG [itex]a>b[/itex].
Since [itex]a[/itex] and [itex]b[/itex] are real numbers, they can be written in binary notation in such a way that each of them has an infinite number of zeroes to the right of the binary (or is it still decimal?) point.
Now, since [itex]a \neq b[/itex] there is a first digit where they differ. At that position, [itex]a[/itex] will have a 1, and [itex]b[/itex] will have a zero. Now, since there is an infinite number of zeros, there is a next location where b has a zero. Let's say that that's the [itex]2^{-k}[/itex]s place.
Then the integer part of
[tex]b*2^{k}+1[/tex]
divided by
[tex]2^{k}[/itex]
Is between [itex]a[/itex] and [itex]b[/itex]