*All sequences considered are non-negative.
Suppose we are given sequences $a_n,b_n$ such that $\sum a_n=1$ and $b_n$ is a sequence going to infinity. I am searching for any sequence $c_n$ with the properties that
(1) $c_n\to \infty$
(2) $c_n\le b_n$
(3) $c_n\cdot\max\lbrace a_k:b_n\le k\le 2b_n\rbrace\to 0$
Does such a sequence $c_n$ exists?
The main difficulty I am having is due to the fact that $na_n$ need not go to zero, so $b_n \max\lbrace a_k:b_n\le k\le 2b_n\rbrace$ need not go to zero (otherwise I would just take $c_n=b_n)$. I am not sure if it is true that there is a $c_n$ (e.g., $c_n= \log b_n$) that satisfies (3).
I have tried to consider $c_n=b_n^a$ with $0<a<1$, however I am not sure if (even for small $a$) we have $b_n^a\max\lbrace a_k:b_n\le k\le 2b_n\rbrace\to 0$. Perhaps the choice of $c_n$ should depend on $a_n$ as well.