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  "body": "\n## Proof\n\n1. Let $s_N = \\sum_{n=0}^{N} a_n$ be the [[partial sums|0NB643K60Q]]. Since $s_{N+1} - s_N = a_{N+1} \\ge 0$, the sequence $(s_N)$ is increasing, as a [[monotone sequence|0KN22AAW5R]].\n2. Assume that $(s_N)$ is bounded above. By the [[monotone convergence theorem|0FM6N4R0VB]], $(s_N)$ converges, and its limit is $\\sup_N s_N$.\n3. By the definition of a [[convergent series|0CVVKY7GY2]], $\\sum a_n$ converges. Its [[sum|0K0H5E3Z5N]] is the limit of $(s_N)$, which is $\\sup_N s_N$ by step 2.\n4. Conversely, assume that $\\sum a_n$ converges. Then $(s_N)$ is a convergent sequence.\n5. [[Convergent sequences are bounded|0S4JF619V7]], so $(s_N)$ is bounded, and in particular bounded above.\n6. Steps 2 to 5 prove the equivalence. When it holds, step 3 gives $\\sum_{n=0}^{\\infty} a_n = \\sup_N s_N$.\n\n## Idea\n\nNonnegative terms make the partial sums increase. An increasing sequence has only one way to fail to converge: it grows without bound. So boundedness is exactly what is needed.\n",
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