Chapter 2 Countable and Uncountable Sets
Equivalence and Cardinality
Pg 18
We say that two sets A and B are equivalent (A~B) if there is a one-to-one correspondence
between them.
Equivalence you need:
(1)one-to-one and
(2)onto.
A function f from A to B is
Limits and Continuity
Pg 14
Punctured Neighborhood of a where we let f be a real-valued function defined (at least) for
all points in some open interval containing the point a R except, possibly, at a itself.
Pg 14 (Limit)
Given a number L R, we write to
Chapter 1 Calculus Review
Pg 3
Of greatest importance to us is that the set R of real numbers is complete - in more than one
sense! First, recall that a subset A of IR is said to be bounded above if there is some x IR such
that a x for all a A. Any such n
Chapter 4 Differentiation
Section 4.1 The Algebra of Derivatives
Pg 69, Definition Let I be an open interval containing the point xo. Then the function is said to be differentiable
a t x o if
exists, in which case we denote this limit by and call it the d
Pg 43, Example 3.1 For each number x, define . Then the function is continuous. To verify this, we select a point
x0 in R, and we will show that the function is continuous at x0. Let be a sequence that converges to x0. By the sum
and product properties of
Chapter 3 Continuous Functions and Limits
Section 3.1 Continuity
Pg 43, Definition A function is said to be continuous at the point xo in D provided that whenever is a sequence in
D that converges to xo, the image sequence converges to f(xo). The function
Pg 25, Proposition 2.1 The sequence cfw_1/n converges to 0; that is,
Pf:
Let . Then by the definition of convergence there is a natural number N such that
Therefore, when . By Corollary 1.10, we can say that there is a natural number N such that . So,
The
Chapter 2 Sequences of Real Numbers
Section 2.1 The Convergence of Sequences
Pg 23, Definition A sequence of real numbers is a real valued function whose domain is the set of natural
numbers.
Pg 25, Definition A sequence is said to converge to the number
Chapter 1 The Real Numbers
Section 1.1 The Completeness Axiom: The Natural, Rational, and Irrational Numbers
Pg 6, Definition A set S of real numbers is said to be inductive provided that
(i) the number 1 is in S
and
(ii) if the number x is in S, the numb
Pg 7, Proposition 1.1 There is no rational number whose square equals 2.
Pf:
For purposes of contradiction we consider that the proposition is false. Suppose there is a rational number x
such that . Since x is rational, we will let , where m and n are int
M TH 511 - R EAL A NALYSIS I
O REGON S TATE U NIVERSITY
FALL 2011
T EXT: R EAL A NALYSIS , N.L. Carothers
L ECTURE N OTES
Chapter 3
Metrics and Norms
Let M be a set and let M M = cfw_(x, y ) : x, y M .
Denition: A function d : M M R is a metric on M if an