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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]], let $(x_n)_{n \\ge 1}$ be a sequence of points of $X$, and let $x \\in X$.\nThe sequence $(x_n)$ converges to $x$ when for each real $\\varepsilon \u003e 0$ there is an integer $N$ such that $d(x_n, x) \u003c \\varepsilon$ for all $n \\ge N$.\nThen $x$ is a limit of $(x_n)$, with the notation $x_n \\to x$ or $\\lim_{n \\to \\infty} x_n = x$.\n\n## Remarks\n\nEquivalently, $x_n \\to x$ if and only if $d(x_n, x) \\to 0$ in $\\mathbb{R}$.\nA sequence in a metric space has at most one limit. This follows from the triangle inequality, and also from the fact that [[limits are unique in Hausdorff spaces|0D8T984XEE]].\nFor $X = \\mathbb{R}$ with the usual metric, this is the usual limit of a real sequence.\n\n## Explanations\n\n### L1 human\n\nA sequence converges to $x$ when its terms come as close to $x$ as you want and stay close.\nFor each tolerance $\\varepsilon$, only finitely many terms are farther than $\\varepsilon$ from $x$.\n",
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