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  "body": "\n## Statement\n\nLet $(a_n)$ be a real sequence.\nIf the series $\\sum a_n$ [[converges absolutely|0W3K3QPEWT]], then $\\sum a_n$ converges.\nMoreover, $\\left| \\sum_{n=0}^{\\infty} a_n \\right| \\le \\sum_{n=0}^{\\infty} |a_n|$.\n\n## Remarks\n\nThe proof uses the [[Cauchy criterion for series|06PVXMY0TH]]. For $n \u003e m$ the triangle inequality gives $\\left| \\sum_{k=m+1}^{n} a_k \\right| \\le \\sum_{k=m+1}^{n} |a_k|$.\nThe converse fails: the [[alternating harmonic series|0JQ5Y3WWVE]] converges, but $\\sum 1/n$ diverges.\nThe result holds in every complete normed space. Mathlib states it as `Summable.of_norm`.\n\n## Explanations\n\n### L1 human\n\nCancellation between terms of opposite signs can only make partial sums smaller in absolute value.\nSo if the series converges without any cancellation, it also converges with it.\n",
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