Convergence along an ultrafilter

Summary

Convergence along an ultrafilter is the filter limit of a function along an ultrafilter. Its key feature is that every function into a compact space has such a limit, and the limit is unique when the space is Hausdorff. So every bounded real sequence has a limit along a fixed non-principal ultrafilter on $\N$, and this limit is linear and multiplicative. Ultrafilter limits are used to build ultraproducts, Banach-limit-like functionals and the Stone–Čech compactification.