Mathematical Analysis 1

Abstract: Preliminary version of a course in univariate real analysis, with 14 chapters and 1 appendix (Ch1--Ch8 complete at present). 1. Infinite sums. Real numbers; 2. Limits of sequences and subsequences; 3. Arithmetic of limits. AK series; 4. Infinite series. Elementary functions; 5. Limits of functions. Asymptotic notation; 6. Continuous functions; 7. Derivatives; 8. Applications of mean value theorems; 9. Taylor polynomials and series. Real analytic functions; 10. Primitives of uniformly continuous functions; 11. Newton integral. Primitives of rational functions; 12. Riemann integral. Transcendence of the number e; 13. Riemann integral. Henstock--Kurzweil integral; 14. More applications of Riemann integral; and A. Solutions to exercises.

• The paper presents a rigorous construction of the real numbers via Cauchy sequences and proofs establishing the uniqueness of the complete ordered field.
• The document employs explicit set-theoretic and logical formalism to power robust limit theory, convergence results, and the detailed treatment of subsequences and infinite series.
• The treatise introduces innovative AK series and combinatorial applications, establishing new algebraic structures with profound implications in analysis and discrete mathematics.

Mathematical Analysis 1: Foundations, Structures, and Innovations

Scope and Structure of the Text

"Mathematical Analysis 1" (2508.19405) is a comprehensive treatise on elementary mathematical analysis, designed as both a rigorous textbook and a reference for foundational concepts in analysis. The work is structured into fourteen chapters, each corresponding to a lecture in the author's course at Charles University, and covers the construction of the real numbers, limits, infinite series, elementary functions, and the algebraic and topological underpinnings of analysis. The text is notable for its systematic approach, explicit set-theoretic formalism, and the integration of exercises and combinatorial applications.

Set-Theoretic Foundations and Logical Formalism

A distinguishing feature of the text is its explicit use of set-theoretic and logical notation throughout. Definitions of sets, classes, relations, and functions are given in terms of ZFC and GB set theory, with careful attention to the axiomatics (extensionality, comprehension, choice, induction, foundation). The treatment of ordered pairs and $k$-tuples is nonstandard, favoring a definition that ensures faithfulness and avoids ambiguities present in iterated pair constructions. Functions are defined as triples $(A, B, G_f)$, where $G_f$ is a functional relation, and congruence of functions is established via equality of graphs, allowing for rigorous manipulation of function spaces and operations.

The logical apparatus is made explicit, with quantifiers, connectives, and tautologies systematically introduced. This formalism is extended to the definition of properties of sequences and robustness, which is crucial for the subsequent development of limit theory and convergence.

Construction and Uniqueness of the Real Numbers

The real numbers are constructed via equivalence classes of Cauchy sequences of rationals (Cantor's construction), with detailed proofs of the correctness of arithmetic operations and order. The completeness of the real numbers is established via the supremum property, and the uniqueness of the complete ordered field is proven by explicit construction of isomorphisms between any two such fields. The text also provides alternative representations of reals via infinite decimal expansions and continued fractions, with bijective correspondences and analysis of multiple expansions for decimal fractions.

The treatment of countability and uncountability is rigorous, with Cantor's theorem and diagonalization arguments used to establish the uncountability of $\mathbb{R}$ and related sets. The embedding of $\mathbb{Q}$ into any ordered field is constructed explicitly, and the Archimedean property is shown to be a consequence of completeness.

Limits, Subsequences, and Robustness

The theory of limits is developed in the context of the extended real line $R^* = \mathbb{R} \cup \{-\infty, +\infty\}$, with precise definitions of neighborhoods, finite and infinite limits, and the arithmetic of limits. The text emphasizes the robustness of properties under finite modifications, and provides strong versions of classical theorems (e.g., monotone convergence, Bolzano–Weierstrass, Cauchy criterion) with robust generalizations.

Subsequences are treated via supports and orderings of subsets of $\mathbb{N}$, and dualities are established: a sequence fails to converge if and only if it has two subsequences with distinct limits. The text proves that any sequence can be partitioned into infinitely many subsequences with a common limit, and provides strong results on partitions and their implications for convergence.

Infinite Series, AK Series, and Algebraic Structures

Infinite series are introduced both in the classical sense and via the concept of AK (absolutely convergent) series, which generalize finite sums to countable index sets and guarantee commutativity and associativity independent of order and grouping. The sum of an AK series is defined as the limit of partial sums over any bijection from $\mathbb{N}$ to the index set, and the class of AK series is shown to form a semiring under binary sum and product operations, modulo congruence.

The text provides detailed proofs of the commutativity and associativity of AK series, and establishes the distributive law in the semiring of factorized AK series. The treatment of infinite series includes the root and ratio tests, Cauchy product, and the analysis of Riemannian and Leibnizian series, with explicit constructions showing that Riemannian series can be reordered to yield any prescribed sum or divergence.

Combinatorial Applications and Fekete's Lemma

A notable innovation is the integration of combinatorial applications of Fekete's lemma, both in additive and multiplicative forms. The text applies these results to extremal functions of words, Szemerédi's theorem, self-avoiding walks in vertex-transitive graphs, meander growth constants, and pattern-avoiding permutations. The existence and computation of growth constants are established via subadditivity and submultiplicativity, and connections to deep results in combinatorics and statistical physics are made explicit.

Elementary Functions and Canonical Forms

Elementary functions are defined rigorously as the closure of basic functions under addition, multiplication, division, and composition. The text introduces canonical forms for polynomials and rational functions, and proves the closure of certain subclasses of elementary functions under differentiation. The exponential and trigonometric identities are derived via Cauchy products, and Euler's formula is proven in the context of complex analysis.

Implications and Future Directions

The formalism and explicitness of the text make it suitable for rigorous analysis, formal verification, and foundational research in mathematical analysis. The AK series framework provides a robust algebraic structure for infinite summation, with potential applications in functional analysis, measure theory, and combinatorics. The integration of combinatorial methods and the emphasis on robustness suggest avenues for further research in the interplay between analysis and discrete mathematics.

The text leaves open the question of whether all elementary functions are closed under differentiation, and points to future work in the extension of the theory to more general spaces, including metric and topological spaces, and the formalization of truth and logical foundations.

Conclusion

"Mathematical Analysis 1" presents a rigorous, explicit, and innovative approach to the foundations of analysis, integrating set-theoretic formalism, robust limit theory, algebraic structures for infinite series, and combinatorial applications. Its systematic treatment and strong results provide a solid basis for further research in analysis, algebra, and combinatorics, and its methods are well-suited for formalization and computational implementation in mathematical software and proof assistants.