Theorem of Heine-Borel

The theorem of Heine-Borel in analysis states that every bounded sequence of real numbers contains a convergent subsequence.

The sequence a₁, a₂, a₃, ... is called bounded if there exists a number L such that the absolute value |a_n| is less than L for every index n. A subsequence is a sequence which omits some members, for instance a₂, a₅, a₁₃, ...

Here is a sketch of the proof: start with a finite interval which contains all the a_n. Cut it into two halves. At least one half must contain a_n for infinitely many n. Then continue with that half and cut it into two halves, etc. This process constructs a sequence of intervals whose common element is limit of a subsequence.

The theorem may also be formulated as:

A subset of the real numbers is compact if and only if it is closed and bounded.

This is true not only for the real numbers, but also for some other metric spaces: the complex numbers, the p-adic numbers, and Euclidean space. However, it fails for the rational numbers and for infinite dimensional normed vector spaces.

The proper generalization is:

A subset of a metric space is compact if and only if it is complete and totally bounded.

The terms complete and totally bounded are explained in the metric space article.