**MATH 3141 - Schedule**

Day | Topics/events | Sections in text |

8/29 | Overview of continuity, differentiability, and integrability. Pitfalls of exchanging limits. | N/A |

9/1 | Field axioms, countability of rationals, rigorous definition of limit. | 1.1, 1.2 |

9/5 | Properties of limits, monotone sequence property, completeness. | 1.2 |

9/7 | Density of rationals (in R), infimum/supremum. | 1.2, 1.3 |

9/12 | Cauchy sequences. | 1.4 |

9/14 | Cluster points, limsup and liminf. | 1.5 |

9/19 | (n-dimensional) Euclidean space, norms, metrics, inner-products. Cauchy-Schwarz inequality. | 1.6, 1.7 |

9/21 | General norm, metric, and inner-product spaces. Open sets. | 1.7, 2.1 |

9/26 | More on open sets, definition of a closed set. | 2.1, 2.3 |

9/28 | Accumulation points and interior points, interior and closure of a set. | 2.2, 2.4, 2.5 |

10/3 | Boundaries, limits and closures. | 2.6, 2.7, 2.8 |

10/5 | Series, Cauchy criteria, comparison theorems for series. | 2.9 |

10/10 | Root and rato tests (their successes and failures). | 2.9 |

10/12 | Test #1. | 1.1-1.7, 2.1-2.6 |

10/17 | Summation by parts, Alternating series. | N/A (bonus material! Though bits in 2.9) |

10/19 | Rearrangements of series. | N/A (more bonus material!) |

10/24 | Compactness: definition and Bolzano-Weierstrauss. | 3.1 |

10/26 | Compactness: Heine-Borel, total boundedness, nested set property. | 3.2, 3.3 |

10/31 | Perfect sets and Cantor sets. | N/A (but see the exercises for Ch. 3) |

11/2 | Continuity, images/pre-images of open, closed, and compact sets. | 4.1, 4.2, 4.3 |

11/7 | Boundedess and extremes of continuous functions on compact sets, uniform continuity. | 4.4, 4.6 |

11/9 | Contunuity and connectedness, intermediate value theorem. | 3.4, 3.5, 4.5 |

11/14 | Test #2. | 2.7-4.1, 4.3 |