Assume that $(a_n)_{n \geq 0}$ and $(b_{n})_{n\geq 0}$ are non-negative sequences satisfying:
- $\sum_{n =0}^{+\infty}a_{n} = +\infty$.
- $(a_{n})_{n\geq 0}$ is decreasing and $\lim_{n\to +\infty}a_{n} = 0$.
- $\sum_{n=0}^{+\infty}a_{n}b_{n} < +\infty$.
Then it can be shown that $\liminf b_{n} = 0$. However, this proof does not need the second assumption.
My question is as follows. Can we get a stronger conclusion, say $\lim b_{n} = 0$, in my setting?
Any help would be appreciated.
No, for example $a_n=1/n$ and $b_n=1$ if $n$ is a square, otherwise $b_n=0$. Then all three conditions are satisfied, but $\lim b_n$ does not exist.