It's a general fact that any series satisfying the list of conditions in http://calculus.subwiki.org/wiki/Alternating_series_theorem http://calculus.subwiki.org

It's a general fact that any series satisfying the list of conditions in http://calculus.subwiki.org/wiki/Alternating_series_theorem http://calculus.subwiki.org/wiki/Alternating_series_theorem is bounded by any two consecutive partial sums; the author probably expects everyone to have seen the proof of this before