Each bounded sequence of real (or complex) numbers contains a convergent subsequence. It can be generalized to include more general objects, e.g. any bounded infinite set in $n$-dimensional Euclidean space has at least one limit point in that space. There exist analogues of this theorem for even more general spaces.
The theorem was demonstrated by B. Bolzano [Bo]; it was later also independently deduced by K. Weierstrass.
|[Bo]||B. Bolzano, Abhandlungen der königlichen böhmischen Gesellschaft der Wissenschaften. v.|
Bolzano-Weierstrass theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Bolzano-Weierstrass_theorem&oldid=32120