Real Analysis I: An Introduction to Theory and Applications (Kindle Bachelor’s Degree in Mathematics)

An Introduction to Theory and Applications

Do you want to master real analysis from a rigorous perspective, with a logical structure and both classical and modern applications in pure and applied mathematics?

This volume, part of the Bachelor in Mathematics series, is a progressive guide that takes the reader from the axiomatic construction of the real numbers to advanced techniques of integration and convergence of functions, combining deep theory, illustrative exercises, and challenging problems.

🔍 What will you find in this book?

✅ Foundations of Real Numbers
Explore the structure of ordered and Archimedean fields, the supremum axiom, the density of rationals and irrationals, and the nested interval principle, with rigorous proofs and applied exercises.

✅ Sequences and Numerical Series
Study the asymptotic behavior of sequences, convergence criteria, rearrangements, and absolutely and conditionally convergent series, with key examples such as the geometric and harmonic series.

✅ Topology of the Real Line
Understand the concepts of open, closed, compact, and connected sets. Apply the Heine–Borel theorem and study adherent and accumulation points, and their role in functional analysis.

✅ Continuity and Limits of Functions
Master the $varepsilon$–$delta$ definition, one-sided and infinite limits, and the theorems of Bolzano, Weierstrass, and Darboux. Analyze continuous functions on intervals and compact sets, with applications to optimization.

✅ Differentiation and Classical Theorems
Apply the derivative as a limit operator, explore Rolle’s theorem and the mean value theorem, develop Taylor series, and study concavity, local extrema, and inflection points.

✅ Integration and Functional Convergence
Introduce the Riemann integral and its properties, integrability criteria, integration techniques, and the analysis of improper integrals. Explore sequences of functions, uniform convergence, and power series.

🧠 Includes:
✔ Step-by-step formal proofs
✔ Solved and proposed exercises by level
✔ Integration of rigorous theory with applications
✔ Historical perspective and axiomatic foundations

📌 Ideal for:

Students of Mathematics and scientific fields

Teachers seeking a rigorous and pedagogical exposition

Researchers wishing to strengthen their analytical foundations

📘 Series: Bachelor in Mathematics – Kindle
🆔 Legal Deposit: 2025-10772
📍 Country of origin: Peru
🏠 Author & Editor: Helbert Justo Luque Zevallos
🔗 Available on Amazon: https://www.amazon.com

Ready to build a solid foundation in real analysis with depth and clarity?