## Math 4433## (Intro. to Analysis I)## Summer 2007 |
Cope's Gray Treefrog This image, copyrighted by Bill Beatty, was taken from www.enature.com, where you can also find a recording of the frog's trilling call. |

Here is a review sheet for the third exam.

The text for this course is currently on reserve in the Mathematics Library, which is located on the 2nd floor of the Physical Sciences Center.

The syllabus for this course is here.

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 | Wed. June 4 | 2.3.5, 2.3.6 |

2 | Thurs. June 5 | 2.3.10, 2.3.11 |

3 | Fri. June 6 | 2.4.1 |

4 | Mon. June 9 | 3.1.4 |

5 | Wed. June 11 | 3.1.8, 3.1.10 |

6 | Thurs. June 12 | 3.1.16 |

7 | Fri. June 13 | 3.1.15 |

8 | Mon. June 16 | 3.2.2(a), 3.2.7 |

9 | Tue. June 17 | 3.2.1(d), 3.2.6(b), 3.2.7 |

10 | Wed. June 18 | 3.2.4, 3.2.18(c,d) |

11 | Wed. June 25 | 3.3.1, 3.3.4 |

12 | Thurs. June 26 | 3.3.9, 3.3.11 |

13 | Fri. June 27 | 3.4.7(b), 3.4.8(b) |

14 | Mon. June 30 | 3.4.10, 3.4.12 |

15 | Tue. July 1 | 3.6.1, 3.6.3 |

16 | Mon. July 7 | 3.7.8, 3.7.14(a) |

17 | Thurs. July 10 | 4.1.8, 4.1.14 |

18 | Tues. July 15 | 4.2.5, 4.2.12 |

19 | Wed. July 16 | 5.1.5, 5.1.11 |

20 | Thurs. July 17 | 5.1.12, 5.1.13 |

21 | Fri. July 18 | 5.2.7, 5.2.8 |

22 | Mon. July 21 | 5.3.1, 5.3.3 |

23 | Tue. July 12 | 6.1.1(b), 6.1.2 |

24 | Wed. July 13 | 6.1.4, 6.1.10 |

- M.I.T. Open Courseware: ocw.mit.edu. Click on "Mathematics" and then "Analysis I, Fall 2006".
- Wikipedia entry for "set theory": en.wikipedia.org/wiki/Set_theory.
- There are many interesting articles on set theory and foundations of mathematics at the online Stanford Encyclopedia of Philosophy.
- "A Primer for Logic and Proof", by H. P. Hirst and J. L. Hirst: www.mathsci.appstate.edu/~jlh/primer/hirst.pdf.
- Here is a site where fundamental theorems in many fields of mathematics are proved in complete detail, with proofs that are verifiable (and have been verified) by computer: us.metamath.org.
- "Understanding Analysis", a textbook by Stephen Abbott.
- "Analysis, Vol. 1", a textbook by Terence Tao.

July 1: The divergent series that I referred to in class today that turns out to be useful in physics is actually 1 + 2 + 3 + ... = -1/12. You can find an explanation of its importance (which requires a "nodding acquaintance with quantum field theory") at John Baez' website: http://math.ucr.edu/home/baez/week126.html. Another place to start which features maybe a gentler introduction is this session of the "Everything" seminar.

For a fairly good account of the arguments that have arisen on the internet concerning the equality 0.999... = 1, see the Wikipedia article titled 0.999....

July 7: The continuum hypothesis states that there is no set that is both "larger in size" than the set of natural numbers and "smaller in size" elements than the set of real numbers. It has been shown that the axioms which are currently accepted as the logical basis for the real numbers do not suffice to either prove or disprove the continuum hypothesis. For more on this topic, including a definition of what exactly "larger in size" and "smaller in size" mean when comparing two infinite sets, see the Wikipedia article titled Continuum hypothesis.

July 15: In class we showed that there exists a function ("Thomae's function") which is discontinuous at all the rational numbers in (0,1) and continuous at all the irrational
numbers in (0,1). This raises the question of whether there exists a function that is continuous at all the rational numbers and discontinuous
at all the irrational numbers. The answer is that no such function can exist. This was first proved by the 20-year old Vito
Volterra in 1881, using a simple argument which you can find in "A Historical Gem from Vito Volterra" by William Dunham,
*Mathematics Magazine*, Vol. 63, No. 4 (Oct., 1990), pp. 234-237. This journal is in the Math library; or, if you are
logged on to an OU computer, you can find the article online at
http://www.jstor.org/stable/2690942. By the way, there are
lots of other interesting facts about calculus and analysis in William Dunham's book, "The Calculus Gallery: Masterpieces from Newton to Lebesgue",
which is available in the Math Library.

July 17: I mentioned the Brouwer fixed point theorem in class: if you start with a piece of rubber occupying a rectangle, crumple it up and stretch
it as much as you like, and then smash it down flat within the original rectangle, there will be at least one point on the piece of rubber which
returns to its original position. There are several proofs of this theorem which rely on sophisticated notions of topology (see the
Wikipedia article on "Brouwer fixed point theorem" for more details) but
a more elementary proof can be given which relates the theorem to the fact that in the simple game of "Hex", at least one player must win.
See "The Game of Hex and the Brouwer Fixed-Point Theorem" by David Gale, in *The American Mathematical Monthly*, Vol. 86, No. 10 (Dec., 1979),
pp. 818-827 (available online through OU servers at http://www.jstor.org/stable/2320146).
You should also try playing Hex yourself; it's fun!
You can play against a computer at http://www.mazeworks.com/hex7/index.htm.