Let's say I have a sequence $a_n \ge 0$ such that I know:
$b \log n - C \le \sum_{i=1}^n a_i \le b \log n + C$
for some constants $b$ and $C$ larger than 0.
How can I prove that:
$a_n = \frac{b}{n} + o(1)\ ?$
This intuitively seems correct because we know that for the harmonic series we get $\sum_{i=1}^n \frac{1}{i} = \log n + o(1)$, but I am not completely sure how to show the reverse.