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

Monday (09/30/96)

SUMMARY Monotone Sequences READING §9 \underline{Warm-Up} (5-10 minutes) Let’s review questions on today’s homework

\underline{Definitions} Let (s_{n}) be a sequence of real numbers. Nondecreasing If s_{n} ≤ s_{n+1} for all n then (s_{n}) is called a nondecreasing sequence Nonincreasing If s_{n} ≥ s_{n+1} for all n then (s_{n}) is called a nonincreasing sequence Monotonic A sequence which is nondecreasing or nonincreasing is called a monotonic se-

quence or monontone sequence \underline{Examples} Let’s do some classifcation of sequences Write down a sequence and determine whether it is

(a) BOUNDED

(b) NONDECREASING

(c) NONINCREASING

(d) CONVERGENT

(e) DIVERGENT

(f) LIMIT D.N.E.

(g) UNBOUNDED SOME USEFUL THEOREMS 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} = −∞.

We like monotone sequences because if (s_{n}) is montonic then lim s_{n} is always “meaningful”, i.e. lim s_{n} exists.

Homework Assignment No Homework due this week: Exam 1 on Friday Oct 4