Bolzano-Weierstrass theorem

The Bolzano-Weierstrass theorem is a statement about the convergence of sequences, and is a basic theorem of real analysis in mathematics.

The Bolzano-Weierstrass theorem holds that if a sequence of real numbers is finite with endpoints, then it must have a subsequence that converges. The theorem holds that an equilibrium exists in the form of a convergent subsequence. The proof of the existence of such an equilibrium has become useful to economics.