Preface; Contents; 1. Sets, Functions, and Real Numbers; 1.1 Sets; 1.1.1 Set Notations; 1.1.2 Subsets; 1.1.3 Operations on Sets; 1.1.4 Exercises; 1.2 Functions; 1.2.1 Functional Notations; 1.2.2 Special Functions; 1.2.3 Inverse Functions; 1.2.4 Composition; 1.2.5 Exercises; 1.3 Real Numbers; 1.3.1 Ordered Field; 1.3.2 Absolute Value; 1.3.3 Upper and Lower Bounds; 1.3.4 The Completeness Axiom; 1.3.5 Lots of Rationals and Irrationals; 1.3.6 Intervals; 1.3.7 Exercises; 1.4 Mathematical Induction; 1.4.1 Induction; 1.4.2 Strong Induction; 1.4.3 Exercises; 1.5 Countability.

1.5.1 Finite and infinite sets1.5.2 Countable and Uncountable Sets; 1.5.3 Exercises; 1.6 Additional Exercises; 2. Sequences; 2.1 Sequences of Real Numbers; 2.1.1 Definition of a Sequence; 2.1.2 Convergent Sequences; 2.1.3 Divergent Sequences; 2.1.4 Uniqueness of Limits; 2.1.5 Exercises; 2.2 Properties of Limits; 2.2.1 Bounded Sequences; 2.2.2 Monotone Sequences; 2.2.3 Arithmetics of Sequences; 2.2.4 Excercises; 2.3 The Bolzano-Weierstrass Theorem; 2.3.1 Subsequences; 2.3.2 Monotone Subsequences; 2.3.3 Cauchy Sequences; 2.3.4 Exercises; 2.4 Limit Superior and Limit Inferior.

2.4.1 Subsequential Limits2.4.2 Size of Subsequential Limits; 2.4.3 Limit Superior and Limit Inferior are Subsequential Limits; 2.4.4 Convergence in Terms of Limit Superior and Limit Inferior; 2.4.5 Exercises; 2.5 Additional Exercises; 3. Series; 3.1 Convergence of Series; 3.1.1 Definition of a Series; 3.1.2 Cauchy Convergence Criterion for Series; 3.1.3 Harmonic and Geometric Series; 3.1.4 Cauchy Condensation Test; 3.1.5 Exercises; 3.2 Comparison Tests; 3.2.1 Comparison Test; 3.2.2 Limit Comparison Test; 3.2.3 Exercises; 3.3 Alternating Series Test; 3.3.1 Exercises; 3.4 Absolute Convergence.

3.4.1 Absolute and Conditional Convergence3.4.2 Root Test; 3.4.3 Ratio Test; 3.4.4 Exercises; 3.5 Rearrangement of Series; 3.5.1 Exercises; 3.6 Additional Exercises; 4. Continuous Functions; 4.1 Limit Points; 4.1.1 Exercises; 4.2 Limits of Functions; 4.2.1 Limits; 4.2.2 Uniqueness of Limits; 4.2.3

Characterization; 4.2.4 One-Sided Limits; 4.2.5 Exercises; 4.3 Continuity; 4.3.1 Sequential Definition of Continuity; 4.3.2

Characterization of Continuity; 4.3.3 Exercises; 4.4 Extreme and Intermediate Value Theorems; 4.4.1 Exercises; 4.5 Uniform Continuity; 4.5.1 Exercises.

4.6 Monotone and Inverse Functions4.6.1 Monotone Functions; 4.6.2 Continuity of Inverse Functions; 4.6.3 Points of Discontinuity; 4.6.4 Exercises; 4.7 Functions of Bounded Variation; 4.7.1 Variation; 4.7.2 Variations on Different Intervals; 4.7.3 Characterization; 4.7.4 Exercises; 4.8 Additional Exercises; 5. Differentiation; 5.1 The Derivative; 5.1.1 Definition of Derivative; 5.1.2 Differentiability and Continuity; 5.1.3 Arithmetics of Derivatives; 5.1.4 Chain Rule; 5.1.5 Derivatives of Inverse Functions; 5.1.6 Exercises; 5.2 Mean Value Theorem; 5.2.1 Two Preliminary Theorems.