Mattuck A. Introduction to Analysis

Pearson (Prentice-Hall division), 1999. — 460 p. — ISBN: 0130811327.The book was developed at MIT, mostly for students not in mathematics having trouble with the usual real-analysis course. It has been used at large state universities and small colleges, as well as for independent study. Students evaluate it as readable and helpful. The current printing, by CreateSpace and at a reduced price, is the eighth, incorporating all known significant corrections and a new Appendix F.
This book is meant for those who have studied one-variable calculus (and maybe higher-level courses as well), generally skipping the proofs in favor of learning the techniques and solving problems. Now they are interested in learning to read proofs, and to find and write up ther own: perhaps because they will need this for the next steps in their chosen field, or for intellectual satisfaction, or just out of curiosity.Real Numbers and Monotone Sequences
Estimations and Approximations
The Limit of a Sequence
The Error Term
Limit Theorems for Sequences
The Completeness Property
Infinite Series
Power Series
Functions of One Variable
Local and Global Behavior
Continuity and Limits of Functions
The Intermediate Value Theorem
Continuous Functions on Compact Intervals
Differentiation: Local Properties
Differentiation: Global Properties
Linearization and Convexity
Taylor Approximation
Integrability
The Riemann Integral
Derivatives and Integrals
Improper Integrals
Sequences and Series of Functions
Infinite Sets and the Lebesgue Integral
Continuous Functions on the Plane
Point-sets in the Plane
Integrals with a Parameter
Differentiating Improper Integrals
Appendix