Ok, so lets say we have these cases

a is a real number between 1 and 2. (non-inclusive)

b is an integer greater or equal to 2.

Now we have a series that is like

floor(a), floor(2a), floor(3a), floor(4a)...where it has infientlely many terms.

Prove that the series will also contain infientely many integer powers of b. That is for any value of a and b that satisfy the cases.