Schedule and Homework

Week	Date	Topics	Homework	Due date
1	May 31^st	Ch 1,8,2	Ch1,12; Ch2,1,10; Ch8,1	June 6^th
	June 1^st	Logic, Ch 3,4	Ch3,1; Ch4,1; Problem1
	June 2^nd	Ch 5	Problem2,3,4
2	June 6^th	Ch 5(continued)	Ch5,1,8; Problem 5,6	June 13^th
	June 7^th	Ch 6	Ch6,1,3
	June 8^th	Ch 7,8,8 Appendix, Exercises	Ch7,11
	June 9^th	Ch 9	Ch9,1,7,10
3	June 13^th	Ch 10	Ch10,1,2,4,5	June 21^th
	June 14^th	Ch 10(continued)	Ch10,8,11
	June 15^th	Ch 11	Ch11,1,2,3,25
	June 16^th	Ch 11(continued), Review	No Homework
4	June 20^th	Midterm	No Homework	June 27^th
	June 21^st	Exercises	No Homework
	June 22^nd	Ch 13	Ch13,9; Problem 7,8
	June 23^rd	Ch 14	Ch14,1,4,5,11
5	June 27^th	Ch 15,18,19	Ch15,1,2; Ch18,1; Ch19,1	Not Collected
	June 28^th	Quiz, Course Review	Ch19,2,3
	June 29^th	Quiz, Course Review(continued)
	June 30^th	Final Exam

Additional assignments:

Problem1:

What is the contrapositive of the following statement?

1. The “well-ordering principle”.

2. For any real numbers a, b, c, if ab=ac, then either a=0 or b=c.

3. For any real numbers a, b, if a, b>0, then ab>0.

Explaining the following statements by method of negation.

1. A is not bounded below.

2. x is not an upper bound of A.

3. x is not a least upper bound of A.

Problem2:

Write the formal definition of “the function f approaches the limit L near a” for ten times.

Problem3:

Prove that the function f(x)=sqrt(x,3) (cubic root of x) approaches 0 near 0.

Problem4:

Prove that the function f(x)=|x| approaches 1 near 1.

Problem5:

Prove that if lim_{x->a}g(x)=m, lim_{x->m}f(x)=L=f(m), then lim_{x->a}f(g(x))=L.

Problem6:

Prove the following by the formal definition of “the function f approaches the limit L near a”:

(1) lim_{x->a}x^n=a^n, where n is an integer;

(2) lim_{x->a}sqrt(x)=sqrt(a), where a>0. (sqrt(a) means the positive square root of a.)

Problem7:

Prove Theorem7 in Chapter13.

Problem8:

Let f(x) be a function defined on [0,2], whose value is 1 at 1 and 0 elsewhere.

Prove that f is integrable on [0,2], and its integration over [0,2] is 0.