Theorem of Bolzano-Weierstrass

The theorem of Bolzano-Weierstrasss 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 is closely related to the theorem of Heine-Borel.