MA 450 Real Analysis

MidTerm

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: 75 minutes

1.

a. What is the definition of a Cauchy sequence?

See definition 18.9, page 158

b. Give an example of a Cauchy Sequence and explain why it is a Cauchy sequence.

By Theorem 18.12, any convergent sequence is a Cauchy sequence. Just give one.

c. Can a divergent sequence be a Cauchy sequence? Explain.

By Theorem 18.12, a sequence is a Cauchy sequence if, and only if, it converges. Thus the answer is NO.

2.

3. .

a. What is the definition of the boundary of S (bdry(S))?

See definition 13.3, page 116.

b.

c. What is the definition of “S is a closed set”?

see definition 13.6, page 117

d. What is the definition of “S is open”?

see definition 13.6, page 117

e.

g.

4.

a.

see definition 12.5, page 107.

b. .

5.

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.