A Basic Course in Real Analysis

Content: Real Number System Algebra of the Real Number System  Upper and Lower Bounds  LUB Property and Its Applications  Absolute Value and Triangle Inequality   Sequences and Their Convergence  Sequences and Their Convergence  Cauchy Sequences  Monotone Sequences  Sandwich Lemma  Some Important Limits  Sequences Diverging to  Subsequences  Sequences Defined Recursively   Continuity  Continuous Functions  Definition of Continuity  Intermediate Value Theorem . Extreme Value Theorem  Monotone Functions  Limits  Uniform Continuity  Continuous Extensions   Differentiation  Differentiability of Functions  Mean Value Theorems  L'Hospital's Rules  Higher-order Derivatives  Taylor's Theorem  Convex Functions  Cauchy's Form of the Remainder   Infinite Series  Convergence of an Infinite Series  Abel's Summation by Parts  Rearrangements of an Infinite Series  Cauchy Product of Two Infinite Series   Riemann Integration  Darboux Integrability  Properties of the Integral  Fundamental Theorems of Calculus  Mean Value Theorems for Integrals  Integral Form of the Remainder  Riemann's Original Definition  Sum of an Infinite Series as an Integral  Logarithmic and Exponential Functions  Improper Riemann Integrals   Sequences and Series of Functions  Pointwise Convergence  Uniform Convergence  Consequences of Uniform Convergence  Series of Functions  Power Series  Taylor Series of a Smooth Function  Binomial Series  Weierstrass Approximation Theorem   A Quantifiers  B Limit Inferior and Limit Superior  C Topics for Student Seminars  D Hints for Selected Exercises  Bibliography  Index