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 (10/07/96)

SUMMARY Introduction to lim sup and lim inf

READING Pages 44-45

\underline{Warm-Up} (5-10 minutes) Review questions on the exam

\underline{Definitions} Let (s_{n}) be a sequence of real numbers. We define

lim sup s_{n} = lim_{N→∞}[sup s_{n} : n > N]

and

lim inf s_{n} = lim_{N→∞}[inf s_{n} : n > N]

That’s nice; but what do these definitions mean?

Maybe this discussion will help (also read page 44 of Ross): Consider the expression in the square brackets. Really, sup s_{n} : n > N is an element of a new sequence. The book calls these terms v_{N}. As N is chosen to be larger and larger the terms v_{N} will get larger and larger and therefor one can find out what their limit is and call this number,lim_{N→∞} v_{N} = v. This number v is known as the lim sups_{n}

A similar phenomena occurs if one thinks about the sequence consisting of inf s_{n} : n > N and the limit of this sequence is called lim inf s_{n}

We shall be using “the limit superior” and “the limit inferior” later on in the course as we discuss a major theorem about the Real Numbers. (so this topic may seem weird but yes you will need to understand it later on in the course also)

\underline{Example}

Consider the sequence s_{n} = (−1)^{n} + \frac{1}{n}

(a) What is lim s_{n}?

(b) Is (s_{n}) bounded? monotone? convergent? divergent?

(c) What is lim inf s_{n}?

(d) What is lim sup s_{n}?

Homework Assignment (Due Monday October 14) §10, page 55: 11.2