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  "body": "\n## Summary\n\nConvergence along an [[ultrafilter|J64Y3JC0RE]] is the [[filter limit|F6QDHKE6J3]] of a function along an ultrafilter.\nIts key feature is that every function into a [[compact|0TVJZYB6SH]] space has such a limit, and the limit is unique when the space is [[Hausdorff|0S1FJ8VEX9]].\nSo every bounded real sequence has a limit along a fixed non-principal ultrafilter on $\\N$, and this limit is linear and multiplicative.\nUltrafilter limits are used to build ultraproducts, Banach-limit-like functionals and the Stone–Čech compactification.\n\n## Explanations\n\n### L1 human\n\nAn ultrafilter on $\\N$ decides, for every set of indices, whether it is \"large\" or \"small\", and exactly one of a set and its complement is large.\nA sequence converges to $x$ along the ultrafilter if, for each small region around $x$, the indices whose terms land there form a large set.\nBecause the ultrafilter takes a side on every set, a bounded sequence always has such a limit.\nFor the sequence $1, 0, 1, 0, \\dots$, indexed from $1$, the limit is $1$ if the set of odd indices is large and $0$ if the set of even indices is large.\n",
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