Cours d'analyse de l'École Royale Polytechnique; Première Partie. Analyse algébrique¶
Cauchy, A. -. (1821). Cours d'analyse de l'École Royale Polytechnique; Première Partie. Analyse algébrique: Analyse algébrique.
Cited by¶
4 citations across 4 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Completeness
- … **Completeness is the no-gaps-in-the-structure principle that names the condition under which a system's internal processes (convergence, deduction, coverage, the construction of canonical extensions) have their natural terminations inside the system itself rather than escaping to a larger ambient structure.**
This sourceFoundational early formulation of the limit-and-continuity framework for real-valued functions; states the Cauchy convergence criterion (the originating articulation of the no-gaps idea in analysis) that anticipates the later epsilon-delta condition, though the criterion was made fully rigorous only with the Cantor/Dedekind completion of 1872.
- … **Completeness is the no-gaps-in-the-structure principle that names the condition under which a system's internal processes (convergence, deduction, coverage, the construction of canonical extensions) have their natural terminations inside the system itself rather than escaping to a larger ambient structure.**
- Continuity
- … is the no-sudden-jumps principle: a mapping or process for which arbitrarily small changes in input produce arbitrarily small changes in output, formally captured by the epsilon-delta condition for real-valued functions (`∀ε > 0, ∃δ > 0 : |x - x₀| < δ ⟹ |f(x) - f(x₀)| < ε`) due to Cauchy (1821) and Weierstrass (1872)
This sourceFoundational early formulation of the limit-and-continuity framework for real-valued functions; gives the first rigorous limit-based definition of continuity and convergence, anticipating the later Weierstrassian epsilon-delta condition.
- … is the no-sudden-jumps principle: a mapping or process for which arbitrarily small changes in input produce arbitrarily small changes in output, formally captured by the epsilon-delta condition for real-valued functions (`∀ε > 0, ∃δ > 0 : |x - x₀| < δ ⟹ |f(x) - f(x₀)| < ε`) due to Cauchy (1821) and Weierstrass (1872)
- Convergence
- **Convergence is the limit-approach principle: a sequence or process is said to converge when its elements eventually enter and remain within every neighborhood of a target limit, formally captured by the epsilon-N condition for sequences (`∀ε > 0, ∃N : n ≥ N ⟹ d(xₙ, x) < ε`)
This source(Originating treatment of the modern sequence-convergence framework, including the Cauchy criterion — a sequence converges in `ℝ` iff it is Cauchy — and the basic theory of convergence of series. Same source publication as the continuity citation; the convergence treatment is distinct enough to warrant a separate inline marker, but the bibliographic entry consolidates in B3 verification.)
- **Convergence is the limit-approach principle: a sequence or process is said to converge when its elements eventually enter and remain within every neighborhood of a target limit, formally captured by the epsilon-N condition for sequences (`∀ε > 0, ∃N : n ≥ N ⟹ d(xₙ, x) < ε`)
- Infinity
- The rigorous formalization of limits in analysis, pioneered by Cauchy (1821)
This source(Foundational early formulation of the limit-and-continuity framework for real-valued functions; introduces the limit-based definition of continuity that anticipates the later Weierstrassian epsilon-delta condition.)
- The rigorous formalization of limits in analysis, pioneered by Cauchy (1821)
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