In mathematics, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space
The theorem states that each infinite bounded sequence in
is sequentially compact if and only if it is closed and bounded.
[3] The Bolzano–Weierstrass theorem is named after mathematicians Bernard Bolzano and Karl Weierstrass.
It was actually first proved by Bolzano in 1817 as a lemma in the proof of the intermediate value theorem.
Some fifty years later the result was identified as significant in its own right, and proved again by Weierstrass.
(set of all real numbers), in which case the ordering on
Indeed, we have the following result: Lemma: Every infinite sequence
Proof[4]: Let us call a positive integer-valued index
Suppose first that the sequence has infinitely many peaks, which means there is a subsequence with the following indices
comes after the final peak, which implies the existence of
Repeating this process leads to an infinite non-decreasing subsequence
; by the lemma proven above there exists a monotone subsequence, likewise also bounded.
denotes its index set) has a convergent subsequence if and only if there exists a countable set
has a convergent subsequence and hence there exists a countable set
, by applying the lemma once again there exists a countable set
This reasoning may be applied until we obtain a countable set
There is also an alternative proof of the Bolzano–Weierstrass theorem using nested intervals.
Also, by the nested intervals theorem, which states that if each
is a closed and bounded interval, say with then under the assumption of nesting, the intersection of the
Proof: (sequential compactness implies closed and bounded) Suppose
Proof: (closed and bounded implies sequential compactness) Since
for which every sequence in A has a subsequence converging to an element of
– i.e., the subsets that are sequentially compact in the subspace topology – are precisely the closed and bounded subsets.
This form of the theorem makes especially clear the analogy to the Heine–Borel theorem, which asserts that a subset of
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.
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.