## Math 4433## (Intro. to Analysis I)## Summer 2011 |
Cooper's Hawk |

- Problems and answers from the third exam.
- Review sheet for the third exam.
- Problems and answers from the second exam.
- Review sheet for the second exam.
- Quiz 3 (Monday, June 20) was on the definition of limit of a function.
- Problems on the first exam.
- Solutions to problems on the first exam.
- Review sheet for the first exam.
- Quiz 2 (Friday, June 3) was on the definition of limit of a sequence.
- Quiz 1 (Thursday, May 26) was on the definition of ``upper bound" and ``supremum" and the statement of the Completeness Property of real numbers.
- Syllabus for the course.

Sometimes I post an assignment in advance but change it in class the day before it's due. If you miss a class you should check this web page after class for the final version of the next day's assignment.

## Assignment |
## Due Date |
## Problems |

1 | Tuesday, May 17 | 2.1.15, 2.1.16(c,d) (here is a scan of the homework problems, in case you haven't found a textbook yet.) |

2 | Wednesday, May 18 | 1.2.13, 1.2.14; 2.1.18 (Problems 1.2.13 and 1.2.14 can be found on this scan, and problem 2.1.18 here.) |

3 | Monday, May 23 | 2.2.3, 2.2.5 |

4 | Wednesday, May 25 | 2.3.1, 2.3.2, 2.3.3 |

5 | Friday, May 25 | 2.3.11, 2.4.5 |

6 | Tuesday, May 31 | 2.4.1, 3.1.3(a,d) |

7 | Wednesday, June 1 | 3.1.5(a) |

8 | Thursday, June 2 | 3.1.13, 3.1.16, 3.1.17 |

9 | Friday, June 3 | 3.2.1(a,b,d), 3.2.5(a) |

10 | Monday, June 6 | 3.2.9 |

11 | Thursday, June 9 | 3.2.14(a), 3.2.15, 3.3.1 |

12 | Friday, June 10 | 3.3.3, 3.3.10 |

13 | Monday, June 13 | 3.4.7(a,b) |

14 | Tuesday, June 14 | 3.4.12 |

15 | Wednesday, June 15 | 3.6.3 |

16 | Thursday, June 16 | 3.4.14 |

17 | Friday, June 17 | 3.7.6(a,b) |

18 | Monday, June 20 | 4.1.3 |

19 | Tuesday, June 21 | 4.1.9(a) |

20 | Friday, June 23 | 4.1.15 |

21 | Monday, June 27 | 4.2.11(a,b), 5.1.11 |

22 | Tuesday, June 28 | 5.1.12, 5.1.13 |

23 | Wednesday, June 29 | 4.2.12, 5.2.12 |

24 | Thursday, June 30 | 5.2.8 |

25 | Friday, June 31 | 6.1.1(a), 6.1.2 |

26 | Tuesday, July 5 | 6.1.12 |

27 | Wednesday, July 6 | 6.2.6, 6.2.7 |

Here is a cleaned-up version of the proof which I gave in class of Theorem 6.2.1. (If you compare with the proof given in the text, you'll see that the text's version is much shorter. This is not because the proof in the text is any different; it's just because the text left out the details which I put in the longer version, in the expectation that readers can supply the details for themselves if they feel like it. One of the things which you have to take into account in writing up a mathematical proof is the balance between readability and detail: too many details and the proof becomes hard to read and understand, too few details and it becomes difficult for readers to fill in the gaps.)

In class we mentioned that there do exist additive functions which are not linear, but that they are very crazy functions. That is, if you put any kind of an extra assumption on an additive function saying that it is in any way "reasonable" (for example, that it is continuous, or that it has a limit at some point), then it must be a linear function. Here is another result along those lines, with a simpler proof than the one given in class: http://math.ou.edu/~jalbert/courses/additive_functions_2.pdf . It is due to G. S. Young.

For a comprehensive history of what has been proved about additive functions and their generalizations, see the last three pages of this paper by Green and Gustin.

Here is a link to an article containing fourteen proofs of the fact that the series generated by 1/n^2 has sum pi^2/6.

Here are a couple of handouts on topics that have come up in class.

- Here is a way to define limits of sequences in terms of the notions of infimums and supremums of sets.
- Here is a reprise of our discussion in class on sheep and logic.

The founder of the subject of analysis, as we learn it in this class, was the French mathematician Augustin-Louis Cauchy (1789-1857). Here is a nice article about what Cauchy did, and why.