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 $\displaystyle \{s_{2n-1}\}$ is a decreasing sequence that is bounded below."

Then, $\displaystyle s_{2n}=(a_{1}-a_{2})+(a_{3}-a_{4})+...+(a_{2n-1}-a_{2n})$

$\displaystyle s_{2n}=a_{1}-(a_{2}+a_{3})-...-(a_{2n-2}+a_{2n-1})-a_{2n}<a_{1}$

$\displaystyle s_{2n-1}=a_{1}-(a_{2}+a_{3})-...-(a_{2n-2}-a_{2n-1})$

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