Real Analysis

Math 310  MWF 10:30-11:25am
Fall 1996  127 Fowler

Ron Buckmire ∼ 402 Fowler ∼ 259-2536 ∼ ron@oxy.edu ∼ Mon 1:30-3:30pm Wed & Thu 1:30-2:30pm

Wednesday (10/02/96)

SUMMARY Review for Exam 1 / More on Monotonic Sequences READING §10 \underline{Warm-Up} (5-10 minutes) We’ll review the Mock Exam together

Connections Let’s try and find the connections between our classifications of sequences. Let (s_{n}) be a sequence of real numbers. Consider the following statements about (s_{n}) and write down which of the statements [(a)-(g)] must be true for each numbered statement:

1. (s_{n}) is bounded ⇒

2. (s_{n}) is unbounded ⇒

3. (s_{n}) is convergent ⇒

4. (s_{n}) is montone ⇒

5. (s_{n}) is montone and bounded ⇒

6. (s_{n}) is montone and unbounded ⇒ Write down a sequence and determine whether it is

(a) (s_{n}) is BOUNDED

(b) (s_{n}) is NONDECREASING

(c) (s_{n}) is NONINCREASING

(d) (s_{n}) is CONVERGENT

(e) (s_{n}) is DIVERGENT

(f) (s_{n}) has NO LIMIT D.N.E.

(g) (s_{n}) is UNBOUNDED SOME USEFUL THEOREMS Let’s quickly go through the proofs of these theorems from the previous class Theorem 10.2: If (s_{n}) is a bounded, nondecreasing sequence, then s_{n} converges.

Theorem 10.4: If (s_{n}) is an unbounded, nondecreasing sequence, then lim s_{n} = +∞. If (s_{n}) is an unbounded, nonincreasing sequence, then lim s_{n} = −∞.

Exam 1 Review

• Peano Postulates

• Natural, Integer, Rational and Real Numbers (ℕ, Z, Q, R)

• Algebraic Numbers and Rational Zeroes Theorem

• Field Axioms

• Maximum, Minimum, Lower Bound and Upper Bound

• Infimum and Supremum

• +∞ and −∞

• Completeness Axiom (For a set S ⊂ ℝ which is bounded above, sup S exists)

• Sequences

• Formal Definition of Convergence of a Sequence

• Formal Proofs

• Theorems about Sequences (e.g. convergent sequences are bounded)

• Theorems on Limits of Sequences (e.g. lim a^{n} = 0 if |a| < 1

• Unbounded sequences (infinite limits)

• Theorems on unbounded sequences

• Monotone sequences (i.e. nondecreasing or nonincreasing)

• Theorems on monotone sequences