This is a problem from the book Understanding Analysis. I am struggling understanding to things. First I do not see how the order limit theorem is used, and second how they go from strictly less to less or equal. Can anybody give me a explanation for these two things?
For a given $m$, $s_{mn}-S$ is a sequence indexed by $n$. If $$-\epsilon< s_{mn}-S<\epsilon$$ then, as $n\to \infty$, $s_{mn}\to (r_1+r_2+\dots +r_m)$, where $r_i=\sum_{j=1}^{\infty}a_{ij}$, and, by the Order Limit Theorem, the inequalities hold but they are no more strict: $$-\epsilon\leq (r_1+r_2+\dots +r_m)-S\leq\epsilon.$$
P.S. Order Limit Theorem: if $\exists N$ such that $a \leq b_n\, \forall n\geq N$, and $\lim_{n\to \infty} a_n=A$, $\lim_{n\to \infty} b_n=B$ then $A\leq B$.
Of course we can replace $a \leq b_n$ with the stronger assumption $a< b_n$ and we may conclude again that $A\leq B$.