
Alternating series test
I found a proof of this that states, "the hypothesis of this test guarantee that the subsequence $\displaystyle \{s_{2n}\}$ is an increasing sequence that is bounded above, while the subsequence $\displaystyle \{s_{2n1}\}$ is a decreasing sequence that is bounded below."
Then, $\displaystyle s_{2n}=(a_{1}a_{2})+(a_{3}a_{4})+...+(a_{2n1}a_{2n})$
$\displaystyle s_{2n}=a_{1}(a_{2}+a_{3})...(a_{2n2}+a_{2n1})a_{2n}<a_{1}$
$\displaystyle s_{2n1}=a_{1}(a_{2}+a_{3})...(a_{2n2}a_{2n1})$
I don't understand why the inequality holds in the second line and leads to the 3rd. Why is it guaranteed to be increasing?
Thanks