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  "body": "\n## Statement\n\nLet $(a_n)$ and $(b_n)$ be real sequences with $0 \\le a_n \\le b_n$ for all $n$.\nIf $\\sum b_n$ converges, then $\\sum a_n$ converges, and $\\sum a_n \\le \\sum b_n$.\nIf $\\sum a_n$ diverges, then $\\sum b_n$ diverges.\n\n## Remarks\n\nThe partial sums of $\\sum a_n$ are at most those of $\\sum b_n$. The result follows from [[a series of nonnegative terms converges iff its partial sums are bounded|0MX1HZBBXJ]].\nIt is enough that $0 \\le a_n \\le b_n$ for all $n$ larger than some $N$.\nFor example, $0 \\le 1/(n^2 + 1) \\le 1/n^2$, so $\\sum 1/(n^2 + 1)$ converges.\nMathlib states it for a summable upper bound, in `Summable.of_nonneg_of_le`.\n\n## Explanations\n\n### L1 human\n\nIf a series of nonnegative terms is smaller term by term than a series with a finite sum, it has a finite sum too.\nIf a smaller series already has an infinite sum, the larger one has an infinite sum as well.\n",
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