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 (10/09/96)

SUMMARY Cauchy Sequences

READING pages 46-47

\underline{Warm-Up} (5-10 minutes) Recall the definition of lim inf and lim sup

\underline{Definitions} Let (s_{n}) be a sequence of real numbers. We say (s_{n}) is a Cauchy sequence if

∀ϵ > 0∃N ∋ m > Nandn > N ⇒ |s_{n} − s_{m}| < ϵ

Write down the definition in your own words:

\underline{Groupwork} Form three groups of two and in each group think of a convergent sequence. Use the formal definition of what a Cauchy sequence is to show that your convergent sequence is Cauchy.

When we say that a sequence is Cauchy we have made quite a strong statement about the sequence. [Think about the connections between statements such as “This sequence is Cauchy.” “This sequence is convergent.” “This sequence is bounded.”] \underline{Theorem} 10.9 All convergent sequences are Cauchy sequences.

\underline{Theorem} 10.10 All Cauchy sequences are bounded.

\underline{Theorem} 10.11 A sequence is convergent if and only if it is a Cauchy sequence.

Homework Assignment (Due Monday October 14) §10, page 47-48: 10.1, 10.6 10.10 §11, page 55: 11.2