Each bounded sequence of numbers contains a convergent subsequence. The theorem applies both to real and complex numbers. 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 ; it was later also independently deduced by K. Weierstrass.
|||B. Bolzano, Abhandlungen der königlichen böhmischen Gesellschaft der Wissenschaften. v. (1817)|
Bolzano–Weierstrass theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Bolzano%E2%80%93Weierstrass_theorem&oldid=28589