Introduction to Analysis by Arthur Mattuck helps the student avoid sudden changes in difficulty. Discusses analysis from the start of the book, to avoid unnecessary discussion on real numbers beyond what is immediately needed. This text includes simplified and meaningful proofs.
This text features Exercises and Problems at the end of each chapter as well as Questions at the end of each section with answers at the end of each chapter. There is analysis in a unified way as the mathematics based on inequalities, estimations, and approximations. This book developed slowly through notes starting around 1975.
It's meant as a first introduction to proofs in analysis, drawing upon the reader's background in one-variable calculus; it's not a technical reference work. It has a lot of unnecessary explanations and cautionary warnings, based on what I've seen on students' problem sets. It has some attempts to smooth over some standard early bumps in the path to understanding. It emphasizes estimation and approximation as the basic tools of analysis, rather than concepts from algebra or point-set topology.
Calculus is used freely from the beginning as a source of examples, so studentscan see how the ideas are used. The real numbers are discussed briefly in the first chapter, with most of the emphasis on the completeness property. The aim is to get to interesting things as quickly as possible. Several appendices present extended applications.
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