Find an $a_1, a_2, \dots$ sequence in the interval $[0,\ln 4]$, such that for any $x<y$ positive integers $$|a_x-a_y|\geq \dfrac{1}{y}$$
I know the well known $$a_n=\dfrac{(-1)^{n+1}}{n}$$ sequence, and I tried making new sequences from that, but none of them worked.
I don't think your sequence $a_n$ is contained in your intervall. For example, $a_2=-\frac{1}{2}\notin [0,ln(4)]$.