**Instructor:** Monica
Visan, MS 6167. Email address: visan@math.ucla.edu

**TA:** Abigail
Hickok,
MS 3919. Email address: ahickok@@math.ucla.edu

**Office Hours:** Mon: 9:30-10:30am, Wed: 2:00-3:00pm, or by
appointment.

**Problem session:** Tu: 11:00-11:50am in MS 5138.

**Textbook:** *Principles of Mathematical Analysis,
3rd Edition* by W. Rudin and *Metric Spaces*, Cambridge University Press, by E. T. Copson.

**Midterm: Wednesday, November 6th, in class.**
The midterm counts for 20%
towards the final grade. You can find a practice midterm here.

**Final: Monday, December 9th, 3:00pm-6:00pm**. You have to take the
final
exam
in
order to pass the class. The final exam will count for 60% towards the
final grade. You can find a practice final here.

**Exam rules:**

- Bring student ID to the midterm and the final.
- There will be no make-up exams.
- No calculators, notes, or books will be permitted in any exam.

**Homework:** There will be weekly homework. It is
due on Fridays *in class*. The homework will count for 20%
towards the final grade. Further information is given
below.

- No late homework will be accepted.
- Write your name at the top of the first page.
- Staple your pages!
- The lowest homework score will be omitted.

**Grading:** Homework 20%; Midterm 20%; Final 60%.

Lecture |
Topics |

1 | Basic Logic, Peano Axioms, mathematical induction. |

2 | Mathematical induction, equivalence relations. |

3 | Construction of the rational numbers. Fields. |

4 | Rational numbers form an ordered field. |

5 | Upper and lower bounds, the least upper bound and the greatest lower bound, the least upper bound property. |

6 | The Archimedean property and consequences. |

7 | Construction of the real numbers. |

8 | "Uniqueness" of an ordered field with the least upper bound property. |

9 | Sequences: bounded, monotonic, convergent. Examples. |

10 | Limit Theorems. Sequences diverging to ±∞. |

11 | The Cauchy criterion. |

12 | The limsup and liminf of a sequence. |

13 | The limsup and liminf of a sequence. |

14 | Series: convergent, absolutely convergent. The Cauchy criterion, the comparison and root tests. |

15 | The ratio test, the Abel, Leibnitz, and dyadic criterions. |

16 | Rearrangements. |

17 | Functions: domain, range, image, preimage, injective, surjective, bijective, inverse. |

18 | Cardinality of a set: finite, infinite, countable, at most countable, uncountable. Examples. |

19 | A set is infinite if and only if it is equipotent to one of its proper subsets. The Schroder-Bernstein theorem. |

20 | A countable union of countable sets is countable. Finite and countable cartesian products of countable sets. Examples. |

21 | Metric spaces, examples, the Holder and Minkowski inequalities. |

22 | Interior, adherent, isolated, and accumulation points. Open and closed sets and their properties. |

23 | Subspaces of a metric space. |

24 | Complete metric spaces: definition and characterization. |

25 | Complete metric spaces: examples. Baire's category theorem. |

26 | Separated sets and properties. |

27 | Connected sets: equivalent definitions. |

28 | Connected subsets of the real line. Decomposition of a set into its connected components. |

**Homework problems:**

- Homework 1 is due in class on Friday, October 4.
- Homework 2 is due in class on Friday, October 11.
- Homework 3 is due in class on Friday, October 18.
- Homework 4 is due in class on Friday, October 25.
- Homework 5 is due in class on Friday, November 1.
- Homework 6 is due in class on Friday, November 8.
- Homework 7 is due in class on Friday, November 15.
- Homework 8 is due in class on Friday, November 22.
- Homework 9 is due in class on Monday, December 2.
- Homework 10 is not to be turned in.