Rein analytischer Beweis des Lehrsatzes, daß zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege

Bolzano, B. (1817). Rein analytischer Beweis des Lehrsatzes, daß zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege.

Type

Book

Intellectual base

Review or monograph

Year

1817

Link

https://archive.org/details/reinanalytische00hankgoog

Cited from

mathematics

Cited by

4 citations across 4 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Boundedness
  • The Bolzano-Weierstrass theorem links the two notions: in finite-dimensional Euclidean space, every bounded sequence has a convergent subsequence — so boundedness implies convergence-of-some-subsequence, even when the full sequence does not converge.

    This sourceearliest rigorous treatment of the intermediate-value theorem and the supremum property; uses the lemma that every bounded infinite sequence has a convergent subsequence, anticipating the Bolzano-Weierstrass theorem.

• Completeness
  • Bolzano (1817) anticipated the least-upper-bound principle in his analytic proof of the intermediate value theorem, supplying the first rigorous non-geometric treatment of the supremum-completeness idea.

    Supported in partVerified against the work's full text

    Bolzano's 1817 treatise rejects geometric proofs of the theorem and proves a greatest-bound lemma used in his analytic proof; it makes no 'first' claim.

    “so gibt es alle- mah! eine Größe V, welche die größte derjenigen ist, von denen behauptet werden kann, daß alle kleineren x die Eigeu-”

    Read by an automated reader; not a human review. How support was checked

• Continuity
  • Analytical use: the role the continuity property plays in the analysis is named — intermediate-value reasoning (existence proofs for solutions of equations
• Topology
  • … $\mathbb{R}^n$ (where compact = closed and bounded) and is the topological correlate of the boundedness-and-closure conjunction in metric spaces; compactness underwrites the existence of maxima and minima of continuous real-valued functions (extreme value theorem), the convergence of subsequences (Bolzano-Weierstrass

Verification

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Links previously used in the corpus

Before the registry existed this work was also linked 2 other ways.

• https://eudml.org/doc/296330 ×1
• https://www.sciencedirect.com/science/article/pii/0315086080900361 ×1

Registry ID ref:1bef98a4555f · see in the full table