With a focus on motivation and a historical thread, this text covers numerical analysis, showing the relevance of computer use to the content. It contains exercises at the end of each section.
(Each chapter concludes with Notes.) 1. The Real Number System. Sets and Operations on Sets. Functions. Mathematical Induction. The Least Upper Bound Property. Consequences of the Least Upper Bound Property. Binary and Ternary Expansions. Countable and Uncountable Sets. Miscellaneous Exercises. Supplemental Reading. 2. Sequence Of Real Numbers. Convergent Sequences. Limit Theorems. Monotone Sequences. Subsequences and the Bolzano/Weierstrass Theorem. Limit Superior and Inferior of a Sequence. Cauchy Sequences. Series of Real Numbers. Miscellaneous Exercises. Supplemental Reading. 3. Structure Of Point Sets. Open and Closed Sets. Compact Sets. The Cantor Set. Miscellaneous Exercises. Supplemental Reading. 4. Limits And Continuity. Limit of a Function. Continuous Functions. Uniform Continuity. Monotone Functions and Discontinuities. Miscellaneous. Exercises. Supplemental Reading. 5. Differentiation. The Derivative. The Mean Value Theorem. LHopitals Rule. Newtons Method. Miscellaneous Exercises. Supplemental Reading. 6. The Riemann And Riemann-Stieltjes Integral. The Riemann Integral. Properties of the Riemann Integral. Fundamental Theorem of Calculus. Improper Riemann Integrals. The Riemann-Stieltjes Integral. Numerical Methods. Miscellaneous Exercises. Supplemental Reading. 7. Series of Real Numbers. Convergence Tests. The Dirichlet Test. Absolute and Condition Convergence. Square Summable Sequences. Miscellaneous Exercises. Supplemental Reading. 8. Sequences And Series Of Functions. Pointwise Convergence and Interchange of Limits. Uniform Convergence. Uniform Convergence and Continuity. Uniform Convergence and Integration. Uniform Convergence and Differentiation. The Weierstrass Approximation Theorem. Power Series Expansion. The Gamma Function. Miscellaneous Exercises. Supplemental Reading. 9. Orthogonal Functions And Fourier Series. Orthogonal Functions. Completeness and Parsevals Equality. Trigonometric and Fourier Series. Convergence in the Mean of Fourier Series. Pointwise Convergence of Fourier Series. Miscellaneous Exercises. Supplemental Reading. 10. Lebesgue Measure And Integration. Introduction to Measure. Measure of Open Sets; Compact Sets. Inner and Outer Measure; Measurable Sets. Properties of Measurable Sets. Measurable Functions. The Lebesgue Integral of a Bounded Function. Lebesgues Theorem. The General Lebesgue Integral. Square Integrable Functions. Miscellaneous Exercises. Supplemental Reading. Bibliography. Hints and Solutions to Selected Exercises. Notation Index. Index.