A Concise Approach to Mathematical Analysis

Numbers and Functions.- Real Numbers.- Subsets of ?.- Variables and Functions.- Sequences.- Definition of a Sequence.- Convergence and Limits.- Subsequences.- Upper and Lower Limits.- Cauchy Criterion.- 3. Series.- Infinite Series.- Conditional Convergence.- Comparison Tests.- Root and Ratio Tests.- Further Tests.- 4. Limits and Continuity.- Limits of Functions.- Continuity of Functions.- Properties of Continuous Functions.- Uniform Continuity.- Differentiation.- Derivatives.- Mean Value Theorem.- L'Hôspital's Rule.- Inverse Function Theorems.- Taylor's Theorem.- Elements of Integration.- Step Functions.- Riemann Integral.- Functions of Bounded Variation.- Riemann-Stieltjes Integral.- Sequences and Series of Functions.- Sequences of Functions.- Series of Functions.- Power Series.- Taylor Series.- Local Structure on the Real Line.- Open and Closed Sets in ?.- Neighborhoods and Interior Points.- Closure Point and Closure.- Completeness and Compactness.- Continuous Functions.- Global Continuity.- Functions Continuous on a Compact Set.- Stone—Weierstrass Theorem.- Fixed-point Theorem.- Ascoli-Arzelà Theorem.- to the Lebesgue Integral.- Null Sets.- Lebesgue Integral.- Improper Integral.- Important Inequalities.- Elements of Fourier Analysis.- Fourier Series.- Convergent Trigonometric Series.- Convergence in 2-mean.- Pointwise Convergence.- A. Appendix.- A.1 Theorems and Proofs.- A.2 Set Notations.- A.3 Cantor's Ternary Set.- A.4 Bernstein's Approximation Theorem.- B. Hints for Selected Exercises.