Welcome to math 260 for fall 2014.
  • Math 260 is a course of central importance. In it you will learn the careful mathematics which justifies the intuitive arguments of calculus. You will come to grips with the epsilons and deltas you flirted with in calculus and find out the huge difference between the notions of continuity and differentiability. You will learn exactly what is meant by a real number and go into depth about the basic properties of the real number system. You will also learn how to write careful proofs and strive for elegance in your writing of them.
  • All this does not come easily and it is a course you will find demanding, probably more demanding than any other course you ever take.
  • The text for the course will be Gaughan "Introduction to analysis" Fifth edition. The book is well written but does not go into as much depth as I would like. I include below a link to the notes for this course last time I gave it, at Berkeley. The notes were taken by a student in the course and are thus a bit raw but should be valuable.

    Notes from previous course.

  • The class is too large for email to be practicable. Please communicate with me at office hours or briefly before and after lectures.
  • Homework is probably more important for this course than for any other course and I will assign homework weekly. A lot of the homework problems will require a considerable amount of reflection so it will be hopeless to start on them the night before they are due. In fact what I want you to do is write up your solutions then wait at least 4 days without looking at them. Then read over them again and see if you can understand them yourself before handing them in.
  • The grade in the course will be based on the homework, two midterms and the final which will be weighted approximately as final, 50%, each mid term 20% and the homework 20%. The astute among you will notice that this does not add up to 100%...
  • I was asked to provide a syllabus so here it is:: Math 260. Introduction to analysis. Exisence and uniqueness of the real numbers. Sequences. Metric spaces.Open and closed sets. Continuity, Compactness and connectedness. Cantor sets. Differentiability. Mean value theorem. Taylor's formula. Convergence of series and series of functions. Stone-Weierstrass theorem. Darboux sums and Riemann integrability. I will hold regular office hours, tentatively Monday 2-3 Wednesday 3-4 Friday 2-3 and by appointment. The appointment should be made right before or after the lectures. These office hours must be used appropriately. They are not for me to do the homework for you or to repeat the lecture for you. Let us say that you may ask me a question but only if you have already thought about it for at least half an hour beforehand. Often you will find that simply by making yourself phrase the question correctly you will be able to answer it wihout help.

    First MIDTERM will be on Friday September 26

    HOMEWORK

    first homework, due Monday 25 August

    selected solutions to first homework, due Monday 25 August

    second homework, due Monday 1 September

    Main ideas of selected solutions to second homework

    third homework, due Monday 8 September

    First MIDTERM will be on Friday September 26

    fourth homework, due Monday 15 September

    CONSTRUCTION OF THE REALS FROM THE RATIONALS

    fifth homework, due Monday 22 September

    WHAT THE FIRST MIDTERM MIGHT LOOK LIKE

    sixth homework, due Monday 29 September updated version!!!!

    seventh homework, due Monday 6 October

    Notes on compactness, Heine Borel and sequential compactness

    eighth homework, due Monday 13 October

    Schroeder Bernstein dynamical systems proof

    homework 9 due Monday 20 October

    Second MIDTERM will be on Friday November 7

    Definition of the Cantor set

    Uniform continuity, pointwise and uniform convergence, examples

    In an uncharacteristic bout of magnanimity I am not assigning any homework due Monday 27 October

    homework 9 due Monday 3 November

    assignments for writing up selected homework exercises

    The second midterm, on Friday (7 November) will cover the course material up to and including last Friday(31 October).

    homework 11 due Monday 10 November

    Students' solution of problem describing all open subsets of the reals

    Students' solution of problem on rearrangements of sequences

    homework 12 due Monday 17 November

    homework 13 due Monday 1 December

    A link from Win containing a dialogue about why the reals are necessary

    Students' solution of problem on definitions of connectedness

    Students' solution of problem on automorphisms of the reals.

    homework 14 not due

    metric space concepts

    A nice discussion of the Cantor set found by Sunny (much more info than required in the course)

    Final from a previous time I taught essentially the same course

    Students' solution of problem on expressing reals in binary, page1.

    Students' solution of problem on expressing reals in binary, page2.