The series $\sum_{n=1}^\infty a_n $ such that its sum is bounded. Given that $a_n\geq 0$ for each $n$, can we prove that $a_n=0$ for infinitely many n?
Given that the sum is bounded, I think that each element must be very small. However, I am not sure if we can say that all $a_n=0$ after some large $n$.
A necessary condition is that the limit of $a_n$ is $0$ as $n\to\infty$. It's not needed that the terms themselves are $0$ after some point. Because that would not be a series, but a finite sum.
Necessary means that it is not enough. There are series where $a_n\to 0$ but are not convergent, like $a_n=\frac{1}{n}$, the harmonic series.
Consider the series $$\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\ldots+\frac{1}{n(n+1)}+\ldots$$ The terms are never zero, nevertheless the series converges. The sum is $1$, by the way.
Hope this helps
This is not in general true.
Consider the series $ \sum_{k\geq 1} \frac1{k^2} $.
