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

Class #3 (09/11/96)

SUMMARY Really big numbers and really small numbers

READING §4, §5

\underline{Warm-Up} (5-10 minutes) Show that |a + b + c| ≤ |a| + |b| + |c|

Therefore, show that |∑_{i=1}^{N} x_{i}| ≤ ∑_{i=1}^{N} |x_{i}|

\underline{Definitions} Let S be a nonempty subset of ℝ.

Maximum If S contains an element s_{0} such that s_{0} ≥ s for all s ∈ S then we call s_{0} the maximum of S and write s_{0} =max S

Minimum If S contains an element s_{0} such that s_{0} ≤ s for all s ∈ S then we call s_{0} the maximum of S and write s_{0} = min S

Upper Bound If there exists a number M ∈ ℝ such that s ≤ M for all s ∈ S, then M is called the upper bound of S and the set S is said to be bounded above

Lower Bound If there exists a number m ∈ ℝ such that m ≤ s for all s ∈ S, then m is called the lower bound of S and the set S is said to be bounded below

Bounded A set S is said to be bounded if it is bounded above and bounded below.

Supremum If S is bounded above and has a least upper bound, then we call this number the supremum of S, denoted by sup S or lub S

Infimum If S is bounded below and has a greatest lower bound, then we call this number the infimum of S, denoted by inf S or glb S \underline{Examples} Determine whether the following sets are bounded above and/or bounded below. Give the supremum/infimum if it exists.

• [0, 1]

• 2, 7

• 1 − 1/3^{n} : n ∈ ℕ

• r ∈ ℚ : r < 2
• r ∈ ℚ : r^{2} < 2
• sin(nπ/3) : n ∈ ℕ

The Completeness Axiom Every nonempty subset S of ℝ that is bounded above has a least upper bound, called sup S.

The Archimedes Property If a > 0 and b > 0, then there exists a positive integer n such that na > b

The Denseness of the Rationals If a, b ∈ ℝ and a < b then there exists a rational number r ∈ ℚ such that a < r < b.

The Symbols +∞ and −∞

\underline{Homework} §4, page 19: Pick 10 sets from 4.1 different from ones done in class.