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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\nThe proof uses the identity $a_n b_n - a b = a_n (b_n - b) + b (a_n - a)$. The sequence $(a_n)$ is bounded, because [[convergent sequences are bounded|0S4JF619V7]].\nWith a constant sequence $a_n = c$, it gives $c\\, b_n \\to c\\, b$.\nBy induction, it gives $a_n^k \\to a^k$ for each natural number $k$.\n\n## Explanations\n\n### L1 human\n\nWe may multiply the limits to get the limit of the product.\nFor example, $(1 + 1/n)(2 - 1/n) \\to 1 \\cdot 2 = 2$.\n",
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