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Part I Central Ideas

Preliminary Remarks

0. Basic Building Blocks
1. The Real Numbers
2. Measuring Distances
3. Sets and Limits
4. Continuity
5. Real-Valued Functions
6. Completeness
7. Compactness
8. Connectedness
9. Differentiation of Functions of One Real Variable
10. Iteration and the Contraction Mapping Theorem
11. The Riemann Integral
12. Sequences of Functions
13. Differentiating f: Rn - Rm

Part II Excursions

1. Truth and Provability
2. Number Properties
3. Exponents
4. Sequences in R and Rn
5. Limits of Functions from R to R
6. Doubly Indexed Sequences
7. Subsequences and Convergence
8. Series of Real Numbers
9. Probing the Definition of the Riemann Integral
10. Power Series
11. Everywhere Continuous, Nowhere Differentiable
12. Newton's Method
13. The Implicit Function Theorem
14. Spaces of Continuous Functions
15. Solutions to Differential Equations

Closer and Closer is the ideal first introduction to real analysis for upper-level undergraduate mathematics majors. The text takes students on a guided journey through the often challenging world of analysis, providing them with the tools to solve rigorous problems with ease. The author achieves this with a student-friendly writing style, an active learning approach, and rich examples and problem sets, along with a unique two-part format. Core Chapters open the text and introduce the most important tools used in analysis. The Excursions then round out and complement Core chapters, allowing students to explore new problems on their own. This two part approach provides a flexible, interactive introduction to relevant concepts and allows students to truly understand and retain key material presented throughout the text. Closer and Closer offers an unparalleled introduction to the foundations of this important area of mathematics.