Real Analysis

Real Analysis

A Visual Approach from Field Axioms to Metric Spaces

von Paul L. Dayao

While many real analysis texts prioritize brevity over clarity, this volume focuses on the student’s conceptual journey.

The text begins with a rigorous construction of the Real Number System from the Field and Order Axioms, moving through the topology of R, sequences, and series. It provides a thorough treatment of continuity and differentiation, including the topological characterization of continuity. A key feature is the rigorous development of the Riemann Integral via Darboux sums, leading to the Lebesgue Criterion. The final chapters transition into Pointwise vs. Uniform Convergence and an introduction to Metric Spaces. The book uses a modern layout with color-coded theorem environments and visual proofs to aid learning.

It is designed as a core text for a one-semester Real Analysis course, specifically for students who find standard texts too opaque.

Über den Autor

Paul L. Dayao is a faculty member at the Ateneo de Manila University who brings over a decade of teaching experience to his writing. His research interests lie in Real Analysis and Functional Analysis. He has recently published works on integration theory in the Missouri Journal of Mathematical Sciences (MJMS) and Real Analysis Exchange.

In addition to his research, he is an active textbook author. He has a book forthcoming with De Gruyter, a book Introductory Course in Functional Analysis  with CRC Press/Taylor & Francis, a college calculus textbook forthcoming with C&E Publishing, and a manuscript Linear Algebra: A Coordinate-Free Approach  currently under review at Springer. He is also finalizing his PhD application in Measure Theory at the University of St Andrews. As a dedicated educator, he specializes in creating “teaching-led” mathematical materials that bridge the gap between intuition and rigor.