Theorem: Each bounded real sequence has a convergent subsequence.

Let {*a*_{n}} be a bounded real sequence. Then there is a real *M* such that for each *n*,*a*_{n} is in [-*M*,*M* ], which is Heine-Borel compact. Let *E* be the set of points in {*a*_{n}}. Then *E* is either finite or infinite.

Suppose *E* is finite. Label the points *p*_{1},...,*p*_{k} and associate with each *p*_{j} the set *E*_{j}={*n*: *a*_{n}=*p*_{j} }. The *E*_{j} form a partition of the positive integers. So one of the *E*_{j} must be countable. Label its elements as a strictly increasing sequence {*n*_{k}}. Thus {*a*_{n_k} } is a convergent subsequence of {*a*_{n}}.

Suppose *E* is infinite. Because it is an infinite subset of the compact set [-*M*,*M* ], *E* has a limit point *A* in [-*M*,*M* ]. So each neighborhood of *A* contains a countable subset of *E*; in other words, each neighborhood of *A* contains countably many terms of {*a*_{n}}. Thus there is a convergent subsequence of {*a*_{n}}.