## Math 4433## (Intro. to Analysis I)## Fall 2008 |
Little Blue Stem This image, by Jim Pisarowicz, was taken from http://www.nps.gov/wica/naturescience/grasses-little-blue-stem.htm. |

Solutions to problems on Exam 2.

Solutions to problems on Exam 1.

The syllabus for this course is here.

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.

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 | Fri. Sept. 5 | 2.3.6, 2.3.9, 2.3.10 |

2 | Fri. Sept. 12 | 3.1.4, 3.1.8 (do these two on a separate sheet of paper); 3.1.5(c), 3.1.6(c) |

3 | Fri. Sept. 19 | 3.1.9, 3.1.10, 3.1.15, 3.1.16 |

4 | Fri. Sept. 26 | 3.2.6(d), 3.2.7, 3.2.9 |

5 | Wed. Oct. 1 | 3.3.1 |

6 | Mon. Oct. 13 | 3.3.4, 3.3.11, 3.3.13(a,b,d) |

7 | Mon. Oct. 20 | 3.4.5, 3.4.9, 3.4.12 |

8 | Mon. Oct. 27 | 3.7.4, 3.7.8 |

9 | Wed. Nov. 5 | 4.1.9(b), 4.1.11(a), 4.1.12, 4.1.14 |

10 | Wed. Nov. 12 | 4.2.2(b,d), 4.2.13, 5.1.7, 5.1.12, 5.1.13 |

11 | Wed. Nov. 19 | 5.3.1, 5.3.3 |

12 | Wed. Dec. 3 | 6.1.1, 6.1.2, 6.1.4 |

- M.I.T. Open Courseware: ocw.mit.edu.
- 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.

Nov. 24: Today in class I mentioned a theorem which states that if you start with a piece of rubber occupying a rectangle, crumple it up and stretch it as much as you like (as long as you don't rip it anywhere), 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. This is a consequence of a more general theorem known as the Brouwer Fixed-Point Theorem. There are many different proofs of this theorem. Some are rather sophisticated (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.

Nov. 19: In class today we discussed additive functions, which are functions that satisfy the equation f(x+y)=f(x)+f(y) for all real numbers x and y. We proved that if a function f is additive and is continuous at 0, then f(x) must equal a constant times x for all real numbers x. The proof was a little long, though. It turns out there is a shorter, more elegant proof of even a somewhat stronger theorem: namely that if a function f is additive and is merely bounded on some interval containing zero (not necessarily continuous at 0), then f(x) must equal a constant times x. I've written it up here.

The study of additive functions goes back all the way to Cauchy, who proved that every additive function must be either continuous everywhere or discontinuous everywhere. The more I look into it, the more it seems to me that almost everything we learn in the first semester of real analysis appeared already in Cauchy's analysis course of the 1820's and 30's. You could say he invented the subject single-handedly.

Another topic we mentioned in class today was the Banach-Tarski paradox (the theorem that says, if you accept the axiom of choice as being true, then you can prove that a solid sphere in three-dimensional space can be cut into 7 pieces which can be reassembled to form two spheres, each the same size as the original.) You can find a lot more about this in the Wikipedia article on the Banach-Tarski paradox. I have never read through a proof of this theorem myself, but if you want to try, my guess is that your best bet would be the article "The Banach-Tarski Paradox" by Karl Stromberg in The American Mathematical Monthly, Vol. 86 (1979), pp. 151-161. From a computer in the OU domain you can easily access this article from www.jstor.org.

Oct. 8: On the first day of class, back in August, I mentioned the question of whether the infinite sum 1 + 2 + 3 + 4 + ... can actually be said to have a numerical value. Looking at this question from one standpoint, the answer is no. Namely, if you define a sequence (x_n) by setting x_1 = 1, x_2 = 1 + 2 = 3, x_3 = 1 + 2 + 3 = 6, x_4 = 1 + 2 + 3 + 4 = 10, and so on, then the sequence (x_n) is obviously unbounded, and so cannot converge. But as Euler pointed out in the 1700's, there are other, more or less rigorous, ways that one might try to assign a numerical value to this infinite sum, and using them one arrives at the strange conclusion that its value is -1/12. (For some idea of how Euler came to this conclusion, see this blog page.) Even more surprisingly, this fact turns out to be relevant in some physical theories (see, e.g., this page at John Baez' website, where it's used to show that "bosonic string theory works best in 26-dimensional spacetime"). You can find more references to this and other closely related infinite sums at this Wikipedia article.