i need to prove that:
$$ \forall i,j \in \{1, \_ ,N \} \subset \mathbb{N} \ \exists k \in \mathbb{N} / (r_{2^k}(i) \leq 2^{k-1} \wedge r_{2^k}(j) > 2^{k-1})\vee (r_{2^k}(i) > 2^{k-1} \wedge r_{2^k}(j) \leq 2^{k-1})$$
i thought that the easy way was just to try it just using induction. But i couldn't prove the "if p(n) then p(n+1)" just adding the value N+1 to the set, so i thoguht that maybe i could calculate that $$ k \in \mathbb{N} $$ so i thought that maybe i could take $$ k \in \mathbb{N}/ j-i <2^k$$ as the solution, im not 100% sure if that's the answer but anyway i could't prove it either.
Could you help me to prove one of those(or maybe both?) (and sorry for my bad spelling, english isn't my first language) EDIT: that $$ k \in \mathbb{N}/ j-i <2^k$$ should be the lower k