Math 370
Real Analysis I
Fall 2011
Instructor: Eric Hayashi
Prerequisites: Grades of C or better in MATH 228 and MATH 301.
Text and Materials:
Basic Elements of Real Analysis by Murray H. Protter, Springer ISBN 0-38798479-8
Supplementary Notes by the inst
Notes for Math 370, Spring 2008
Eric Hayashi
San Franciso State Univerity
May 5, 2008
Abstract
1. Real Number System R
1. properties of the real numbers
1.
2.
3.
4.
5.
6.
deciencies with Q
ordered eld properties
upper and lower bounds, inf S , sup S
Axiom
Math 370
Solutions to Assignment 1
5, 8.1, 8.2, 18, 22, 25, 28.
5. Let A = fn 2 N : n > 3g : Then A has no maximal element.
Proof. Suppose that A has a greatest element m: Then m 2 A so 3 < m < m +1 2 A: This contradicts
the property that m is maximmal in
MATH370
Fall 2011
Solutions to HW 2
Exercise 45. Prove that limn!1
1
n2
= 0:
Proof. Let " > 0: By the Archimedean property of the reals, there exists a natural number N such
that N > 1=": Thus 0 < 1=N < ": For every natural nuumber n, we have
n
N)
1
n2
1