Books like Analysis I by Terence Tao

Real and Complex Analysis by Walter Rudin is a rigorous, advanced textbook suitable for a one‑ or two‑semester course in mathematical analysis, used by mathematics, science, computer science, and electrical engineering majors at the junior, senior, or graduate level.

A comprehensive bridge from elementary calculus to advanced real and complex function theory, this textbook introduces the abstract reasoning characteristic of modern mathematical analysis.

A comprehensive introduction to real analysis, suitable for advanced undergraduate mathematics students.

As the title implies, this is an introductory text on mathematical analysis. It focuses on the logical basis of particular math topics which nowadays (as of 2012) are typically featured in a pre-calculus text. The 1967 teacher's edition is accessible to anyone who understands basic algebra. It is designed to prepare students to approach math in a methodical and rigorous manner from an elementary level. Some of the topics are outdated--it includes log and other tables. Although it is an elementary text, the approach used by the authors was meant to introduce logical rigor into high-school mathematics. The lessons are concerned with structure; some of the methods are quite out of favor now that electronic calculators are ubiquitous. This is the sort of math that a student ought to be able to appreciate without a calculator, i.e., it is more concerned with logical structure and proof (at least by the authors' standards) than with memorization of axioms without proof, backed by blind faith in calculators. At the time the text was first written there were no handheld calculators, so elegant algorithms were in demand. The text was designed to teach students how to construct algorithms based on mathematical reasoning. The one exception would be the inclusion of various log, trig, and other tables in the back that were probably computer generated, the algorithms for which were slightly beyond the scope of the text.

A Beginners choice for learning Real Analysis.

An accessible bridge from calculus to real analysis, offering straightforward proofs, motivating explanations, and numerous examples – including new topics such as π’s irrationality, Baire category, and continuous nowhere‑differentiable functions.

Ben shu zhu yao fen san bu fen:di yi bu fen wei shi bian han shu lun, Di er bu fen wei chou xiang kong jian, Di san bu fen wei yi ban ce du yu ji fen lun.

"This text leads the reader through the various tactics involved in solving mathematical problems at the Mathematical Olympiad level. Solving Mathematical Problems includes numerous exercises and model solutions throughout. Assuming only basic high-school mathematics, the text is ideal for general readers and students 14 years and above with an interest in pure mathematics."--P. [4] of cover.

This volume contains techniques of integration which are not found in standard calculus and advanced calculus books. It can be considered as a map to explore many classical approaches to evaluate integrals. It is intended for students and professionals who need to solve integrals or like to solve integrals and yearn to learn more about the various methods they could apply. Undergraduate and graduate students whose studies include mathematical analysis or mathematical physics will strongly benefit from this material. Mathematicians involved in research and teaching in areas related to calculus, advanced calculus and real analysis will find it invaluable.The volume contains numerous solved examples and problems for the reader. These examples can be used in classwork or for home assignments, as well as a supplement to student projects and student research.