Genuine mathematical interest
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An ultrafilter is a filter that is not strictly contained in any other filter. Despite this high degree of abstraction, the theory of ultrafilters is particularly useful in mathematics, either because it provides elegant proofs of difficult theorems or because it enables new fundamental concepts to be created. Yet here lies a paradox: no one can construct a non-trivial ultrafilter. There are even "many" of them, as can be proved using the famous axiom of choice, but it is impossible to exhibit one. Thus, mathematics sometimes uses objects that are assumed to exist but that no one knows how to construct!
Convergence and limits
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The definition of convergence for a sequence u of real numbers (that is, a map from ℕ to ℝ) is well known: u converges to the real number a if, for every ε > 0, there exists an integer N such that, if n ≥ N, then |u(n) ‒ a| < ε.
If E is a subset of ℕ, it is said to be cofinite if its complement Ec is finite. Thus, the set E*ε of integers n such that |u(n) ‒ a| < ε is cofinite if u converges to a. Conversely, if Eε = {n* ℕ: |u(n) ‒ a| < ε} is cofinite for every ε > 0, then the sequence u converges to a.
