Helly's Selection Theorem

Helly's selection theorem, named for the Austrian mathematician Eduard Helly, states that a uniformly bounded sequence of monotone real functions has a convergent subsequence, making it a sequential compactness theorem for the space of uniformly bounded monotone functions. A more general version establishes compactness for the space of functions that are locally of bounded total variation and uniformly bounded at a point, and the theorem has applications throughout analysis, including implying compactness of a tight family of probability measures.