PLEASE: Write up your solutions/proofs in LaTex, Word, etc., and email it to me or bring it to class when it is due. Here is a link to a Latex tutorial (thanks Shoshana!)
http://www.maths.tcd.ie/~dwilkins/LaTexPrimer/
Or you can ask me questions about Latex directly.
I want to see you write a perfect proof of a "simple" fact. I will grade this one! Prove that if $a_n < b_n, n=1,2,...,$ and $a_n \to a, b_n \to b$, then $a \le b$.
section 8.1 #3,4,5
section 8.2 #1,5
section 8.3 #4
section 8.6 #4
section 8.7 #4,5,6,8
Everybody should plan to attend and support Prof. Langou's "career day" exibit on Monday, February 22. It's during our class, so class
is cancelled on that day. If you are curious about careers in math, then you will find people to talk to there. And if you already have a career
that involves math then you should be on the other end of the discussion and let people know what you do.
section 9.1 #1,3,6,8
Justify your conclusions in #1. Use the hint in #6.
Remember: No class on monday 2/22. Please attend "career day" during our normal class time.
It's in room 470, UCD building, from 5:30 to approximately 6:30 (or beyond). There will be pizza!
section 9.2 #1-4, 6
section 9.3 #1-4, 6
(1) Several students made the same mistake on problem #4 in section 8.7 (from homework #2). This is the one where you were supposed to show that the Weierstrass theorem does not hold on bounded open intervals. It's true that the function $f(x) = 1/(b-x)$ is not uniformly continuous on $(a,b)$. It's also true that the proof of Weierstrass' theorem in the book made good use of the fact that a continuous function on a closed bounded set is uniformly continuous. But, you cannot conclude from those facts that it is impossible to uniformly approximate f(x) by a polynomial on (a,b)! In other words, if A implies B, it does not follow that (not A) implies (not B). As "collective punishment" :-) for all those mistakes, I want you all to prove the following: If f(x) is not uniformly continuous on the bounded open interval (a,b), then there exists $/epsilon >0$ such that if p(x) is a polynomial then there exists $x \in (a,b)$ such that $|p(x) - f(x)| > \epsilon$. You will see that this is not exactly "obvious" (although it's true), so you cannot just assume it is true in a proof.
(2) Construct a sequence of functions, $f_n: [0,1] \to \Re$ such that each $f_n$ is Reimann integrable,
the sequence converges pointwise, i.e. $f_n(x) \to f(x), x \in [0,1]$, but $f(x) is not Reimann integrable.
HINT: See if you can think of one where each $f_n$ is zero except at a finite number of points, but f(x) is nonzero
on an infinite set of points.
section 9.4 #1,3,4
section 9.5 #1,3,8
note: The midterm exam will be on wednesday, March 17 (in class)
section 11.1 #5,6,9,11
section 11.2 #1,4
section 12.2 #1,2
Let $(X,d_1)$ and $(X,d_2)$ be metric spaces where $d_1 \equiv d_2$. Prove that $x_n \to x$ in $(X,d_1)$ if and only if $x_n \to x$ in $(X,d_2)$.
section 13.1 #3,11
section 13.2 #5
section 13.3 #4,9