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  "body": "\n## Definition\n\nLet $S$ be a set and let $\\mathcal{U}$ be an [[ultrafilter|J64Y3JC0RE]] on $S$.\nLet $X$ be a [[topological space|0CQHFPGJ45]], let $f \\colon S \\to X$ be a map, and let $x \\in X$.\nThe map $f$ *converges to $x$ along $\\mathcal{U}$* if\n$$\\set{s \\in S : f(s) \\in V} \\in \\mathcal{U}$$\nfor every [[neighbourhood|0ZCY0Z80B6]] $V$ of $x$.\nWhen this limit is unique it is written $\\lim_{\\mathcal{U}} f$ or $\\mathcal{U}\\text{-}\\lim_{s} f(s)$.\n\n## Remarks\n\nThis is the [[filter limit|F6QDHKE6J3]] of $f$ along the filter $\\mathcal{U}$.\n\nExample: for $a \\in S$ and the principal ultrafilter $\\mathcal{U}_a = \\set{A \\subseteq S : a \\in A}$, every $f$ converges to $f(a)$ along $\\mathcal{U}_a$.\nIndeed, for each neighbourhood $V$ of $f(a)$ the set $\\set{s : f(s) \\in V}$ contains $a$.\n\nNon-example: let $\\mathcal{U}$ be a non-principal ultrafilter on $\\N$ and $f(n) = n$ in $\\R$.\nFor any $x \\in \\R$, the set $\\set{n : \\abs{n - x} \u003c 1}$ is finite, and a non-principal ultrafilter contains no finite set.\nSo $f$ converges to no point along $\\mathcal{U}$.\n\nIf $X$ is [[compact|0TVJZYB6SH]], then every $f \\colon S \\to X$ converges along $\\mathcal{U}$ to some point; this is closely related to [[compactness via ultrafilters|DT5PH9QVQ7]].\nIf $X$ is [[Hausdorff|0S1FJ8VEX9]], the limit is unique.\nFor a non-principal ultrafilter $\\mathcal{U}$ on $\\N$, the map $x \\mapsto \\lim_{\\mathcal{U}} x_n$ on bounded real sequences is linear and multiplicative, and its value lies between $\\liminf_n x_n$ and $\\limsup_n x_n$.\nIt is not a [[Banach limit|SYF2ND6J97]]: it gives $(1, 0, 1, 0, \\dots)$ the value $0$ or $1$, while every Banach limit gives $1/2$.\n",
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