1. Introduction to Real Analysis
Dr. Weihu Hong
Clayton State University
10/16/2008

2. Open cover
Definition Let E be a subset of R. A collection of open subsets of R is an open cover of E if
Alternative definition: The collection of open sets is an open cover of E if

3. Compact Sets
Definition Let K be a subset of R. K is compact if every open cover of K has a finite subcover of K; that is, if is an open cover of K, then there exists
Remark. Please pay attention to that it has to be true for every open cover of K.

4. Properties of Compact Sets
Theorem 3.2.5
(a) Every compact subset of R is closed and bounded.
(b) Every closed subset of a compact set is compact.
Remark. Be able to prove these two statements.

5. Properties of Compact Sets
Theorem 3.2.6
If S is an infinite subset of a compact set K, then S has a limit point in K.
Remark. Be able to prove this theorem in two different ways.

6. Properties of Compact Sets
Theorem 3.2.7
If is a sequence of nonempty compact subsets of R with for all n, then
is nonempty and compact.

7. Characterization of the Compact Subsets of R
Theorem (Heine-Borel)
Every closed and bounded interval [a, b] is compact.
Theorem (Heine-Borel-Bolzano-Weierstrass)
Let K be a subset of R. Then TFAE:
(a) K is closed and bounded.
(b) K is compact.
(c) Every infinite subset of K has a limit point in K.
Theorem Let K be a nonempty compact subset of R. Then every sequence in K has a convergent subsequence that converges to a point in K.