Bolzano-Weierstrass theorem

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 [1]; it was later also independently deduced by K. Weierstrass.

References

[1]	B. Bolzano, Abhandlungen der königlichen böhmischen Gesellschaft der Wissenschaften. v. (1817)