= Solution
Let $s_N=\sum_{n=1}^N(-1)^{n-1}b_n$. Pairing consecutive terms shows $s_{2m}\geq0$ and
$$
s_{2m+2}-s_{2m}=b_{2m+1}-b_{2m+2}\geq0,
\qquad s_{2m+3}-s_{2m+1}=-b_{2m+2}+b_{2m+3}\leq0.
$$
The even <partial sums> increase and the odd <partial sums> decrease. They satisfy $s_{2m}\leq s_{2m+1}\leq b_1$, while
$$
s_{2m+1}-s_{2m}=b_{2m+1}\longrightarrow0.
$$
Both are <monotone bounded sequences> and hence have limits, and the displayed difference makes their limits equal. Every partial sum belongs to one subsequence, so \b[the alternating series converges]. This is a proof of the <alternating series test>; the monotonicity and zero-limit hypotheses play separate roles.
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