MA 450 Real Analysis

Final Exam

Notes: You must show all of your work. Each sub- problem has the same weight.

You can use any earlier result in the exam even if you haven't been able to prove the earlier result.

Time: 3 hours

3.

a.

see definition 12.5, page 107.

b. .

5.

a. What is the definition of a Cauchy sequence?

See definition 18.9, page 158

6.

a.

see definition 14.1, page 123.

b. If S is compact prove that S is bounded. (Note: you can’t just cite the Heine-Borel theorem in this problem)

c. State the Heine-Borel Theorem (don’t prove it).

See theorem 14.5, page 124.

e. Prove that any intersection of compact sets is compact.