I'm going to be as vague as possible while trying to provide enough details.
I have a set which is defined by certain conditions, one of them being that for any two values x,y in the set, we have that $\frac{x+y}{2}$ is also in the set.
When I write out the values of the set, I found that each value can be expressed in the form $\frac{m}{2^n}$, where n,m are non-negative integers.
If I wanted to prove that every element in the set can be expressed as such, how would I do so?
I'm assuming it would be inductively, but I'm not sure what the inductive step would be. Showing $n+1$ wouldn't exactly work, because the next value is not necessarily of a denominator $2^{n+1}$.
My next thought is to assume there are two arbitrary values a,b and then put them through the condition above ($\frac{a+b}{2}$), with a and b each expressed in the form $\frac{m}{2^n}$, then show that the result can be expressed in the same form $\frac{m}{2^n}$. Would this work?