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  "body": "\n## Proof\n\nWe apply the Cauchy criterion to the sequence of partial sums.\n\n1. Let $s_n = \\sum_{k=0}^{n} a_k$ be the [[partial sums|0NB643K60Q]]. For natural numbers $n \u003e m$, we have $s_n - s_m = \\sum_{k=m+1}^{n} a_k$.\n2. By definition, $\\sum a_n$ is a [[convergent series|0CVVKY7GY2]] exactly when $(s_n)$ converges.\n3. By the [[Cauchy criterion for sequences|0ZA4AX4MW3]], $(s_n)$ converges if and only if it is a [[Cauchy sequence|0QZMTN5AXX]].\n4. Assume that $(s_n)$ is Cauchy, and let $\\varepsilon \u003e 0$. There is $N$ with $\\abs{s_n - s_m} \u003c \\varepsilon$ for all $m, n \\ge N$. For $n \u003e m \\ge N$, step 1 turns this into $\\abs{\\sum_{k=m+1}^{n} a_k} \u003c \\varepsilon$.\n5. Conversely, assume the condition of the statement, and let $\\varepsilon \u003e 0$. Take $N$ as in the condition, and let $m, n \\ge N$.\n6. If $m = n$, then $\\abs{s_n - s_m} = 0 \u003c \\varepsilon$. If $n \u003e m$, step 1 and the condition give $\\abs{s_n - s_m} \u003c \\varepsilon$. If $m \u003e n$, swap the roles of $m$ and $n$.\n7. So $(s_n)$ is a Cauchy sequence.\n8. By steps 2 to 7, the series converges if and only if the condition holds.\n\n## Idea\n\nA block sum $a_{m+1} + \\cdots + a_n$ is the difference $s_n - s_m$ of two partial sums. So the condition says exactly that the partial sums form a Cauchy sequence, and Cauchy sequences of reals converge.\n",
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