Real Analysis Volume 1 Real Numbers, Sequences and Continuity (Bachelor's Degree in Mathematics)

Real Analysis: Volume 1 - Foundations of the Real Numbers and Sequences is a rigorous, structured guide for students, instructors, and researchers who want to deeply understand the fundamental principles of mathematical analysis. This book offers a progressive journey from the structure of the real numbers to the study of sequences, series, and continuity of functions, with a solid theoretical focus accompanied by detailed exercises.
What will you find in this book?
Structure of the Real Numbers
A rigorous exploration of ordered fields, the least upper bound axiom, and the completeness of the real numbers. Essential concepts such as supremum and infimum, the density of the rationals, and the nested intervals principle are addressed.
Sequences and Convergence Criteria
A formal study of numerical sequences, including convergence, limit superior and limit inferior, monotone sequences, and the Cauchy criterion. Fundamental theorems such as Bolzano-Weierstrass and their applications in real analysis are presented.
Numerical Series and Absolute Convergence
Development of the theory of series, covering classical convergence tests such as comparison, ratio, and root tests. The difference between absolutely and conditionally convergent series and the effect of rearranging terms are studied.
Topology of the Real Line and Continuity of Functions
An introduction to open and closed sets, compactness, and connectedness on the real line. Continuity of functions is studied along with fundamental theorems such as Weierstrass's theorem (extreme value theorem) and the Intermediate Value Theorem.
Differentiability and Applications of the Derivative
A rigorous definition of the derivative, rules of differentiation, and its relationship with local growth of functions. Taylor's formula, concavity analysis, and applications to optimization problems are covered.
Solved and Proposed Exercises
Each chapter includes a broad collection of exercises carefully designed to reinforce understanding and develop advanced mathematical skills.
Highlights:
✔ Clear, detailed explanations: Ideal for learners seeking a structured, progressive path.
✔ Rigorous examples and proofs: Each concept is presented with theorems and applications in mathematical contexts.
✔ Didactic approach: Designed to ease comprehension from fundamentals to advanced results.
Who is this book for?
Students of mathematics, physics, and engineering who need a formal, structured introduction to real analysis.
Instructors and self-learners seeking rigorous material with well-grounded theory and challenging exercises.
Researchers and professionals who require a reference text on sequences, series, and function continuity.
With a balance between theory and practice, Real Analysis: Volume 1 is the perfect resource for university courses and independent study in mathematical analysis.

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