why this statement about $\sum_{i=0}^n a_n$ is false?

Question

Given that ${a_n}$ is positive series, and $\sum_{i=0}^n a_n$ is converge:

-There is a sub-series for ${a_n}$ that converge to S>$0$. (THIS statement is false)

why is this statement false?

Answer 1

Since $$\sum_{i=0}^n a_n$$ is convergent, we have $$\lim _{n\to \infty} a_n =0$$

Therefore any sub-sequence of $\{a_n\}$ also converges to to $0$.

Thus it is not possible for a sub-sequence to converge to a positive number.