We can write where denotes the integral part of
Now, we can say that as
Let where is odd. Then
All such integers, must satisfy the property where is a non-negative integer.
Now, Means, is even.
Or, is an integer.
Hence, is an integer for all positive reals satisfying
for some non-negative integer
As takes infinitely many values, number of such positive real numbers is also infinite.
This completes the proof.