Let $n > 3$ be an integer. Prove that there exist positive integers $x_1,..., x_n$ in geometric progression and positive integers $y_1,..., y_n$ in arithmetic progression such that $x_1<y_1<x_2<y_2<...<x_n<y_n$
It is enough to find positive rational geometric, arithmetic progression
Let $x_m=(1+\epsilon)^m\approx 1+m\epsilon$(but bigger than $1+m\epsilon$),$\epsilon$ is very small rational number. Let $y_m=1+(m+1)\epsilon-\delta$ ($\epsilon>\delta\gg \epsilon^2$,$\delta$ is rational number) then it completes when $\epsilon$ is sufficiently small.
Then by cutting first few terms we know it satisfy for every $n$