Reading guide · Proof index

Selected prerequisite proofs

Jiří Lebl, Basic Analysis: Introduction to Real Analysis, volumes I and II, version 6.3. These 105 exact proof selections accompany the stationary-phase lesson and its local completions.

Used under CC BY-SA 4.0; see the original dual-licence notice and free author edition. The human proofs retain their authorship. GPT-6 Astra (OpenAI), Ultra, arranged this selection, routed references and adapted the typesetting.

Only selected statements, proofs and referenced figures are included. Navigation and seven optional biographical or out-of-scope footnotes are omitted; their source remains unchanged. One proof used by two statements appears once, with its equal-integrals application written at the second use. The other retained formulas and proof prose are preserved. Referenced exercise completions and clarifications are linked to the programme supplements. Notation and definitions provide the common reading context.

• Limit superior, limit inferior, and Bolzano–Weierstrass
• Cauchy sequences
• Complex exponential and trigonometric functions
• Differentiation under the integral
• Facts about limits of sequences
• Fundamental theorem of calculus
• Improper integrals
• Iterated integrals and the Fubini theorem
• Jordan measurable sets
• Limits of functions
• The logarithm and the exponential
• Completeness and compactness
• Continuous functions
• Fixed point theorem and Picard’s theorem again
• Metric spaces
• Sequences and convergence
• Open and closed sets
• Extreme and intermediate value theorems
• More on series
• Change of variables
• Higher order derivatives
• Mean value theorem
• Analysis with vector spaces
• Outer measure and null sets
• The set of Riemann integrable functions
• The Riemann integral
• Properties of the integral
• Riemann integral over rectangles
• Sequences and limits
• Inverse and implicit function theorems
• The derivative
• Continuity and the derivative
• Vector spaces, linear mappings, and convexity