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  "body": "\n## Proof\n\nWe show that the even and the odd [[partial sums|0NB643K60Q]] are monotone and squeeze the sum.\n\n1. For $m \\le n$, $b_n \\le b_m$, since $(b_n)$ is decreasing, as a [[monotone sequence|0KN22AAW5R]]. Since $b_n \\to 0$ and [[limits preserve non-strict inequalities|0N32V0E3XJ]], $0 \\le b_m$ for every $m$.\n2. For each $m$, $s_{2m+2} = s_{2m} - (b_{2m+1} - b_{2m+2}) \\le s_{2m}$. So $(s_{2m})$ is decreasing.\n3. For each $m$, $s_{2m+3} = s_{2m+1} + (b_{2m+2} - b_{2m+3}) \\ge s_{2m+1}$. So $(s_{2m+1})$ is increasing.\n4. For each $m$, $s_{2m+1} = s_{2m} - b_{2m+1} \\le s_{2m}$.\n5. By steps 2 to 4, $s_1 \\le s_{2m+1} \\le s_{2m} \\le s_0$ for every $m$.\n6. By the [[monotone convergence theorem|0FM6N4R0VB]], $(s_{2m+1})$ converges to $O = \\sup_m s_{2m+1}$. Applied to $(-s_{2m})$, it shows that $(s_{2m})$ converges to $E = \\inf_m s_{2m}$.\n7. Since $b_{2m+1} \\to 0$ by [[subsequences of a convergent sequence|0VHAVXW08V]], step 4 and the [[limit of a sum|0RAM4E07NR]] give $O = E - 0 = E$. Call this number $S$.\n8. Let $\\varepsilon \u003e 0$. There is $K$ with $\\abs{s_{2m} - S} \u003c \\varepsilon$ and $\\abs{s_{2m+1} - S} \u003c \\varepsilon$ for all $m \\ge K$. So $\\abs{s_N - S} \u003c \\varepsilon$ for all $N \\ge 2K$.\n9. So $s_N \\to S$, by the definition of the [[limit|0XKWSSY8P2]], and the series converges with sum $S$.\n10. By step 6, $s_{2m+1} \\le S \\le s_{2m}$ for every $m$, because $S$ is the supremum of the odd sums and the infimum of the even sums.\n11. Let $N = 2m$ be even. By step 10, $s_{2m+1} \\le S \\le s_{2m}$. So $\\abs{S - s_N} = s_{2m} - S \\le s_{2m} - s_{2m+1} = b_{2m+1} = b_{N+1}$.\n12. Let $N = 2m + 1$ be odd. By step 10 with $m + 1$, $s_{2m+1} \\le S \\le s_{2m+2}$. So $\\abs{S - s_N} = S - s_{2m+1} \\le s_{2m+2} - s_{2m+1} = b_{2m+2} = b_{N+1}$.\n13. Steps 11 and 12 give $\\abs{S - s_N} \\le b_{N+1}$ for each $N$.\n\n## Idea\n\nThe partial sums zigzag: down by $b_1$, up by $b_2$, down by $b_3$, and so on, with shrinking steps. So the even sums decrease, the odd sums increase, and the two families squeeze a common limit. The limit always lies between two consecutive partial sums, which gives the error bound.\n",
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