Contents
1 Review of Real Analysis 2
1.1 The Real Number System . . . . . . . . . . . . . . . . . . . . 2
1.2 Innite Sequences of Real Numbers . . . . . . . . . . . . . . . 4
1.3 Monotonic Sequences . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Subsequences and the Bolzano-Weierstrass Theorem . . . . . . 8
1.5 Cauchy's Criterion for Convergence . . . . . . . . . . . . . . . 9
1.6 Limits of Functions of a Real Variable . . . . . . . . . . . . . 10
1.7 Continuous Functions of a Real Variable. . . . . . . . . . . . . 13
1.8 The Intermediate Value Theorem . . . . . . . . . . . . . . . . 16
1.9 Continuous Functions on Closed Bounded Intervals . . . . . . 17
2 Convergence, Continuity and Open Sets in Euclidean Spaces 19
2.1 Open Sets in Euclidean Spaces . . . . . . . . . . . . . . . . . . 25
2.2 Closed Sets in a Metric Space . . . . . . . . . . . . . . . . . . 28
2.3 Continuous Functions and Open and Closed Sets . . . . . . . 29
2.4 Continuous Functions on Closed Bounded Sets . . . . . . . . . 30
3 Metric Spaces 32
3.1 Convergence and Continuity in Metric Spaces . . . . . . . . . 33
3.2 Open Sets in Metric Spaces . . . . . . . . . . . . . . . . . . . 35
3.3 Closed Sets in a Metric Space . . . . . . . . . . . . . . . . . . 37
3.4 Continuous Functions and Open and Closed Sets . . . . . . . 39
3.5 Homeomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.6 Complete Metric Spaces . . . . . . . . . . . . . . . . . . . . . 41
3.7 The Completion of a Metric Space . . . . . . . . . . . . . . . . 43