Past exam of the mathematics course of the University of Cambridge / 2013 / ia / Paper 1 / 4F / a

Let $s_N={\sum }_{n=1}^N(-1{)}^{n-1}{b}_n$. Pairing consecutive terms shows $s_{2m}\ge 0$ andThe even partial sums increase and the odd partial sums decrease. They satisfy $s_{2m}\le s_{2m+1}\le b_1$, whileBoth are monotone bounded sequences and hence have limits, and the displayed difference makes their limits equal. Every partial sum belongs to one subsequence, so the alternating series converges. This is a proof of the alternating series test; the monotonicity and zero-limit hypotheses play separate roles.