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  "body": "\n## Statement\n\nLet $(a_n)$ be a [[convergent sequence|0AZ3R52593]] of real numbers.\nThen $(a_n)$ is a [[Cauchy sequence|0QZMTN5AXX]].\n\n## Remarks\n\nLet $L$ be the limit and let $\\varepsilon \u003e 0$. Take $N$ with $|a_n - L| \u003c \\varepsilon/2$ for $n \\ge N$. Then $|a_m - a_n| \\le |a_m - L| + |L - a_n| \u003c \\varepsilon$ for $m, n \\ge N$.\nThe statement holds in each metric space.\nThe converse holds in $\\mathbb{R}$, but it fails in $\\mathbb{Q}$ and in other spaces that are not complete.\n\n## Explanations\n\n### L1 human\n\nIf all late terms are close to one number $L$, then they are also close to each other.\nTwo terms that are each within $\\varepsilon/2$ of $L$ are within $\\varepsilon$ of each other.\n",
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