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  "body": "\n## Statement\n\nLet $(a_n)$ and $(b_n)$ be real sequences, and let $a$ and $b$ be real numbers.\nIf $a_n \\to a$ and $b_n \\to b$ as $n \\to \\infty$, then $a_n + b_n \\to a + b$ as $n \\to \\infty$.\n\n## Remarks\n\nGiven $\\varepsilon \u003e 0$, choose $N$ so that $|a_n - a| \u003c \\varepsilon/2$ and $|b_n - b| \u003c \\varepsilon/2$ for all $n \\ge N$. The triangle inequality then gives $|(a_n + b_n) - (a + b)| \u003c \\varepsilon$.\nWith the constant sequence $-1$ and the [[product rule for limits|0734RSVF2V]], it also gives $a_n - b_n \\to a - b$.\nIt is one of the rules of the [[algebra of limits|07XBM3A9J5]].\n\n## Explanations\n\n### L1 human\n\nIf $a_n$ is close to $a$ and $b_n$ is close to $b$, then $a_n + b_n$ is close to $a + b$.\nThe two errors add, so we make each error smaller than half of the allowed error.\n",
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