What is Helly's selection theorem?

In mathematics, Helly's selection theorem states that a uniformly bounded sequence of monotone real functions admits a convergent subsequence.In other words, it is a sequential compactness theorem for the space of uniformly bounded monotone functions.It is named for the Austrian mathematician Eduard Helly.A more general version of the theorem asserts compactness of the space BVloc of functions locally of bounded total variation that are uniformly bounded at a point.

The theorem has applications throughout mathematical analysis. In probability theory, the result implies compactness of a tight family of measures.