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  "body": "\n## Statement\n\nLet $(a_n)$ be a [[convergent sequence|0AZ3R52593]] of real numbers.\nThen $(a_n)$ is [[bounded|05FQ3VGAR9]]: there is a real number $K \\ge 0$ such that $|a_n| \\le K$ for all $n \\in \\mathbb{N}$.\n\n## Remarks\n\nLet $L$ be the limit. Take $\\varepsilon = 1$ in the [[epsilon-N definition|09G6KHS6GY]] to get $N$. Then $K = \\max(|a_0|, \\dots, |a_{N-1}|, |L| + 1)$ works.\nThe converse is false. The sequence $(-1)^n$ is bounded and divergent.\nThe statement holds in each metric space, where bounded means contained in some ball.\n\n## Explanations\n\n### L1 human\n\nAfter some index, all terms are close to the limit, so they lie in a small interval.\nOnly finitely many terms come before that index, and finitely many numbers are always bounded.\n",
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