douglass-analysis-sol

This document is a solutions manual to accompany Douglass, Steven A. Introduction to Mathematical Analysis. Addison-Wesley, 1996. ("IMA")

IMA was the textbook of the introductory analysis courses—MAS241 Analysis I and MAS242 Analysis II—offered by the Department of Mathematical Sciences in KAIST. This paper is a collection of my solutions to exercises studying introductory analysis with this book.

All solutions are based on the author's definitions, assuming that all theorems, including lemmas and corollaries, and examples covered in the text can be cited without proof. If one quotes a proposition introduced in another exercise, see the solution to that problem for proof. In particular, there are many equivalent propositions about notions of the elementary topology. Please keep that in mind when you solve problems from other sources.

This paper is unofficial and might still contain some errors, although I have worked on minimizing them. You can contribute to it via my email. All reports and suggestions are welcome.

Solved Exercises

1. The Structure of the Real Numbers: Sequences
   1.4, 1.5, 1.6, 1.7, 1.8, 1.12, 1.14, 1.15, 1.16, 1.19, 1.20, 1.23, 1.26, 1.27, 1.29, 1.30, 1.35, 1.39, 1.41, 1.47, 1.50, 1.59, 1.65, 1.66, 1.67, 1.68, 1.69, 1.70, 1.71, 1.73, 1.75, 1.78, 1.79, 1.82, 1.83, 1.84, 1.85, 1.91, 1.93, 1.94, 1.95, 1.96, 1.99, 1.108, 1.109, 1.110
2. Euclidean Spaces
   2.2, 2.3, 2.4, 2.5, 2.7, 2.8, 2.12, 2.15, 2.17, 2.19, 2.20, 2.21, 2.24, 2.25, 2.31, 2.32, 2.33, 2.34, 2.35, 2.36, 2.37, 2.38, 2.41, 2.42, 2.44, 2.45, 2.46, 2.47, 2.48, 2.49, 2.50, 2.51, 2.53, 2.55, 2.58, 2.61, 2.68, 2.69, 2.72, 2.73, 2.74, 2.75
3. Continuity
   3.4, 3.12, 3.13, 3.15, 3.16, 3.17, 3.18, 3.20, 3.21, 3.23, 3.26, 3.28, 3.31, 3.32, 3.33, 3.37, 3.38, 3.39, 3.40, 3.41, 3.42, 3.47, 3.59, 3.60
4. Differentiation
   4.2, 4.6, 4.7, 4.13, 4.14, 4.15, 4.17, 4.20, 4.24, 4.25, 4.26, 4.27, 4.28, 4.30, 4.32, 4.33, 4.34, 4.46, 4.47, 4.50, 4.51, 4.52, 4.53, 4.58, 4.68, 4.69, 4.70
5. Functions of Bounded Variation
   5.3, 5.10, 5.11, 5.12
6. The Riemann Integral
   6.1, 6.2, 6.7, 6.8, 6.9, 6.12, 6.14, 6.21, 6.23, 6.29, 6.33, 6.41, 6.59, 6.63, 6.71, 6.96
7. The Riemann-Stieltjes Integral
   7.22
8. Differential Calculus in $\mathbb{R}^n$
   8.2, 8.13, 8.16, 8.47, 8.49
9. Vector-Valued Functions
   9.12
10. Multiple Integrals
    10.18
11. Infinite Series
    11.10, 11.19, 11.26
12. Series of Functions
    12.6, 12.16