Source and edition
Jiří Lebl, Basic Analysis: Introduction to Real Analysis, volume I, version6.3.
The teaching unit follows the chapters listed below in its own words and is dedicated to the public domain under CC0 1.0. GPT-6 Astra (OpenAI), in Codex, at Ultra, wrote it, including the direct nested-interval and Cauchy argument, elementary operations and derivative rules, finite-subcover and finite-exception details, the Cauchy mean value proof and worked solutions, and self-checked it.
Author’s editable chapters
- ch-real-nums.tex — Complete Archimedean and rational-density proof; preceding real-field existence paragraph also read and explicitly omits its proof.
- ch-seq-ser.tex — Bolzano–Weierstrass bisection proof and Cauchy-completeness section. The programme uses the directly supplied bisection/subsequence route, not an imported limsup theorem.
- ch-contfunc.tex — Extreme values, intermediate values and uniform-continuity proofs; all actual source passages used read.
- ch-der.tex — Fermat, Rolle, ordinary and Cauchy mean value statements and proofs. The source leaves the Cauchy version as an exercise; the programme gives the full proof.
- ch-riemann.tex — Darboux definitions/refinement/inequalities, continuous integrability, both fundamental-theorem proofs. Programme adds elementary operations and full finite-exception proofs.
The unit uses the complete ordered field proved in Constructing the real numbers, Theorems4,5,7 and10. That preceding unit explicitly identifies its rational arithmetic and set-theoretic starting inputs.