Math 199 Schedule

Math 199 Schedule

Monday	Wednesday
	August 24 Syllabus Introduction to the Real numbers
August 29 More on the Real Numbers Vectors, or elements of R^n	August 31 Section 1.3, cont.
September 5 No Classes - Labor day	September 7 Homework 1 due Convex sets Read Section 1.4 Section 1.5 - Topology
September 12 Topology, cont. Section 1.6 - Sequences	September 14 (Really 16 September) Section 1.6, cont
September 19 Problems due: From HW #2 - 2, 4 From HW #3 - 2, 5, 7, 9, 10 Section 1.6, 1.7	September 21 lim sup and lim inf Section 1.8 - compactness
September 26 Compact Sets Homework problems from HW #4 are due	September 28 Heine-Borel Theorem
October 3 Continuity	October 5 Open set definition of continuity Defn of Uniformly continuous
October 10 Uniformly continuous	October 12 PL Approximations Homework due on Friday by noon
October 17 Implications of Continuity	October 19 Implications of continuity Exam on Friday 9 am - 12 noon
October 24 Section 2.5 - Limits of Functions	October 26
October 31 Homework 8 due	November 2
November 7	November 9
November 14	November 16
November 21	November 23 No Class - Thanksgiving Break
November 28	November 30
December 5	December 7

August 24

Topics Covered	Homework
Syllabus Introduction to the real numbers R forms a field R has an ordering R is an ordered field LUBs and GLBs	Homework 1

August 24

Topics Covered	Homework
Syllabus Introduction to the real numbers R forms a field R has an ordering R is an ordered field LUBs and GLBs	Homework 1

August 29

Topics Covered	Homework
Introduction to the real numbers, cont Some sets have upper bounds, but not LUBs Existance of the real numbers Archimedean property Denseness of the rational numbers in the reals R^n is a vector space over R dot products	Homework 1 Also solve: Suppose $S$ is an ordered set with the lub-property, $B \subset S$, $B \neq \emptyset$ and $B$ is bounded below. Let $L$ be the set of all lower bounds of $B$. Then $\alpha=\sup L$ exists in $S$ and $\alpha=\inf B$. In particular, $\inf B$ exists in $S$.

August 31

Topics Covered	Homework
Denseness of the rational numbers in the reals revisited Dot products Properties of dot products Metrics	Homework 1 Also solve: Suppose $S$ is an ordered set with the lub-property, $B \subset S$, $B \neq \emptyset$ and $B$ is bounded below. Let $L$ be the set of all lower bounds of $B$. Then $\alpha=\sup L$ exists in $S$ and $\alpha=\inf B$. In particular, $\inf B$ exists in $S$. The above is due on Wednesday, 7 Sept. Homework 2

September 7

Topics Covered	Homework
Convex sets interior of a set open sets exterior of a set closed sets accumlation points isolated points	Homework 3 Homework 2

September 12

Topics Covered	Homework
Disconnected and Connected Sets Polygon connected Path connected Sequences Trace of a sequence Bounded sequence Convergent sequences Every convergent sequence is bounded.	The Following is due on Monday, 19 Sept. Homework 3 Homework 2

September 14
(Really this was September 16)

Topics Covered	Homework
A set $S$ is closed if and only if it contains the limit of every converging sequence $\{p_n\}$ whose terms lie in $S$.	The Following is due on Monday, 19 Sept. Homework 3 Homework 2 From HW #2 - 2, 4 From HW #3 - 2, 5, 7, 9, 10

September 19

Topics Covered	Homework
Every bounded monotonic sequence is convergent. Nested Interval Theorem Bolzano-Weierstrass Theorem Cauchy Sequences	Homework 4, due Monday, 26 September

September 21

Topics Covered	Homework
lim sup and lim inf Definition of Compact Compact sets in R^k are bounded Compact sets in R^k are closed	Homework 4, due Monday, 26 September Homework 5

Sept 26

Topics Covered	Homework
Compact Sets Heine Borel	Homework 5

September 28

Topics Covered	Homework
Heine-Borel Theorem Proof of the Heine Borel Theorem	Homework 5

October 3

Topics Covered	Homework
Continuity Defn. of continuous sequence defn. of continuity	Homework 6

October 5

open set defn

Uniform Continuity

Defn of uniform continuity
example of a function which is cont, but not uniformly cont.

October 10

Topics Covered	Homework
Uniform Continuity Defn of uniform continuity example of a function which is cont, but not uniformly cont. continuous function on compact set is uniformly cont. uniform continuity preserves Cauchy sequences	Homework 6 (#5, 6, 7) and Homework 7 (#1 and 3) due on Friday 10/14 by noon

October 12

Topics Covered	Homework
PL Approximations Weierstrass Approximation Theorem	Homework 6 (#5, 6, 7) and Homework 7 (#1 and 3) due on Friday by noon

October 17

Topics Covered	Homework
Implications of Continuity local properties global properties locally positive functions Image of a cont. fun. on a compact set is bdd. Extreme Value Theorem cont. image of a compact set is compact pre-image of f(p)>c and f(p)=c	Study for the exam!