{
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  "body": "\n## Definition\n\nLet $(a_n)$ be a [[real sequence|0CTV783HYT]] and let $c$ be a real number.\nThe number $c$ is a cluster point of $(a_n)$ if the following holds for each $\\varepsilon \u003e 0$ and each natural number $N$.\nThere is a natural number $n \\ge N$ such that $|a_n - c| \u003c \\varepsilon$.\n\n## Remarks\n\nEquivalently, for each $\\varepsilon \u003e 0$ there are infinitely many indices $n$ with $|a_n - c| \u003c \\varepsilon$.\nIn $\\mathbb{R}$, the number $c$ is a cluster point of $(a_n)$ if and only if some [[subsequence|0R8J3ZQMJS]] of $(a_n)$ converges to $c$.\nThe sequence $(-1)^n$ has the two cluster points $1$ and $-1$, and no limit.\nFor a bounded sequence, the [[limit superior|0P8BA7T7S5]] is the largest cluster point and the [[limit inferior|0H4R3RESGV]] is the smallest.\n\n## Explanations\n\n### L1 human\n\nA cluster point is a number that the sequence visits again and again, as closely as you like.\nA limit is a cluster point that the sequence approaches with all its late terms, not only with some of them.\n",
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