Introductory Problem Courses in Analysis and Topology

Description

Front Matter....Pages i-vii Front Matter....Pages ix-ix Notations....Pages 1-1 The Real Numbers, Regarded as an Ordered Field....Pages 2-5 Functions, Limits, and Continuity....Pages 6-10 Integers. Sequences. The Induction Principle....Pages 11-14 The Continuity of IR....Pages 15-21 The Riemann Integral of a Bounded Function....Pages 22-27 Necessary and Sufficient Conditions for Integrability....Pages 28-31 Invertible Functions. Arc-length and Path-length....Pages 32-36 Point-wise Convergence and Uniform Convergence....Pages 37-39 Infinite Series....Pages 40-43 Absolute Convergence. Rearrangements of Series....Pages 44-45 Power Series....Pages 46-48 Power Series for Elementary Functions....Pages 49-50 Front Matter....Pages 51-51 Sets and Functions....Pages 53-56 Metric Spaces....Pages 57-59 Neighborhood Spaces and Topological Spaces....Pages 60-64 Cardinality. Finite and Countable Sets....Pages 65-68 The Completeness of IR. Uncountable Sets....Pages 69-70 The Schröder-Bernstein Theorem....Pages 71-71 Compactness in IR n ....Pages 72-75 Front Matter....Pages 51-51 Compactness in Abstract Spaces....Pages 76-77 The Use of Choice in Existence Proofs....Pages 78-80 Linearly Ordered Spaces....Pages 81-82 Mappings Between Metric Spaces....Pages 83-85 Mappings Between Topological Spaces....Pages 86-87 Connectivity....Pages 88-90 Well-ordering....Pages 91-92 The Existence of Well-orderings. Zorn’s Lemma....Pages 93-94 Back Matter....Pages 95-95