Bolzano-Weierstrass theorem

Both proofs involved what is known today as the Bolzano-Weierstrass theorem .

The Bolzano-Weierstrass theorem states that every bounded sequence of real numbers has a convergent subsequence.

The Bolzano-Weierstrass theorem.

Since the sequence (a, p, b, q) lies in this compact set, it must have a convergent subsequence by the Bolzano-Weierstrass theorem.

In the 1880s, it became clear that results similar to the Bolzano-Weierstrass theorem could be formulated for spaces of functions rather than just numbers or geometrical points.

This implies, by the Bolzano-Weierstrass theorem, that any infinite sequence from the set has a subsequence that converges to a point in the set.

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.

Ekeland's variational principle can be used when the lower level set of a minimization problems is not compact, so that the Bolzano-Weierstrass theorem can not be applied.

The Bolzano-Weierstrass theorem gives an equivalent condition for sequential compactness when considering subsets of Euclidean space: a set then is compact if and only if it is closed and bounded.

It can be used to prove many of the fundamental results of real analysis, such as the intermediate value theorem, the Bolzano-Weierstrass theorem, the extreme value theorem, and the Heine-Borel theorem.

Important results include the Bolzano-Weierstrass theorem and Heine-Borel theorems, the intermediate value theorem and mean value theorem, the fundamental theorem of calculus, and the monotone convergence theorem.

A space X is compact if and only if every net (x) in X has a subnet with a limit in X. This can be seen as a generalization of the Bolzano-Weierstrass theorem and Heine-Borel theorem.

Today he is mostly remembered for the Bolzano-Weierstrass theorem, which Karl Weierstrass developed independently and published years after Bolzano's first proof and which was initially called the Weierstrass theorem until Bolzano's earlier work was rediscovered .

Using this definition and the concept of uniform convergence, Weierstrass was able to write proofs of several then-unproven theorems such as the intermediate value theorem (for which Bolzano had given an insufficiently rigorous proof), the Bolzano-Weierstrass theorem, and Heine-Borel theorem.

The culmination of their investigations, the Arzelà-Ascoli theorem, was a generalization of the Bolzano-Weierstrass theorem to families of continuous functions, the precise conclusion of which was that it was possible to extract a uniformly convergent sequence of functions from a suitable family of functions.

These last two properties, together with a lemma used in the proof of the Bolzano-Weierstrass theorem, yield one standard proof of the completeness of the real numbers, closely related to both the Bolzano-Weierstrass theorem and the Heine-Borel theorem.