Theorem: Each bounded real sequence has a convergent subsequence.
Let {an} be a bounded real sequence. Then there is a real M such that for each n,an is in [-M,M ], which is Heine-Borel compact. Let E be the set of points in {an}. Then E is either finite or infinite.
Suppose E is finite. Label the points p1,...,pk and associate with each pj the set Ej={n: an=pj }. The Ej form a partition of the positive integers. So one of the Ej must be countable. Label its elements as a strictly increasing sequence {nk}. Thus {an_k } is a convergent subsequence of {an}.
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 {an}. Thus there is a convergent subsequence of {an}.