Heine–Borel theorem, which asserts that a subset of is compact if and only if it is closed and bounded. In fact, general topology tells us that a metrizable space is compact if and only if it is sequentially compact, so that the Bolzano–Weierstrass and Heine–Borel theorems are essentially the same. Application to economics There are different important equilibrium concepts in economics, the proofs of the existence of which often require variations of the Bolzano–Weierstrass theorem. One example is the existence of a Pareto efficient allocation. An allocation is a matrix of consumption bundles for agents in an economy, and an allocation is Pareto efficient if no change can be made to it that makes no agent worse off and at least one agent better off (here rows of the allocation matrix must be rankable by a preference relation). The Bolzano–Weierstrass theorem allows one to prove that if the set of allocations is compact and non-empty, then the system has a Pareto-efficient allocation. See also Sequentially compact space Heine–Borel theorem Completeness of the real numbers Ekeland's variational principle Notes References External links A proof of the Bolzano–Weierstrass theorem PlanetMath: proof of Bolzano–Weierstrass Theorem The Bolzano-Weierstrass Rap Theorems in real analysis Compactness theorems