does there exist such numbers satisfying this limit condition?

Question

Does there exist $a_n\ge 0$ and $b_n\ge 0$, $n\ge 1$, such that $$\lim_{x\to 0^+} \dfrac{\sum_{x>a_n} b_n}{x} =1$$?

Answer 1

If I am not wrong, and if I have well understood the question, $a_n=\frac{1}{n}$ and $b_n=\frac{1}{n(n+1)}=\frac{1}{n}-\frac{1}{n+1}$ works.