If the sum of any finite terms in a sequnce is less than or equal to epsilon, then prove that the infinite sum is also less than or equal to epsilon.

Question

We have a sequence of sequences which converges to $a_{i}$ i.e. $$ \{ x_{i}^m \}_{m=1}^{\infty} \rightarrow a_{i} \quad \forall \; i \in \mathbb{N} $$

for any $\epsilon >0$ and for any $M$, if we can prove that there exist some $m$ s.t.

$$ \sum_{i=1}^{M} |x_{i}^m - a_{i}| \leq \epsilon $$

then, does it naturally imply or do we have to prove the following result:

$$ \sum_{i=1}^{\infty} |x_{i}^m - a_{i}| \leq \epsilon \quad $$

Answer 1

It is nothing more than the fact that $u_{M}\leq\epsilon$, $M=1,2,...$ implies that $u\leq\epsilon$, where $u=\lim_{M\rightarrow\infty}u_{M}$.