1
Chapter One: The Real Number System
1.1
Lecture 1: Complete Ordered Fields
We will characterize the set R of real numbers as a complete ordered eld
containing the rational numbers. We begin by describing a eld.
Denition 1.1. A eld F is a set with two op
2
2.1
Chapter Two: Sequences
Lecture 1: Convergence and Basic Properties
Denition 2.1. i A sequence cfw_an is a function from the positive integers
to the real numbers; occasionally, the domain is the set of nonnegative
integers.
ii cfw_an converges to
3
3.1
Chapter Three: Limits and Continuity
Lecture 1: Limits
Denition 3.1. i Suppose a R. By a neighborhood N (a) of a is meant
an interval of the form (a r, a + r) for some r > 0. The neighborhood
is also written as Nr (a). A deleted neighborhood of a is
3
3.1
Chapter Three: Limits and Continuity
Lecture 1: Limits
Denition 3.1. i Suppose a R. By a neighborhood N (a) of a is meant
an interval of the form (a r, a + r) for some r > 0. The neighborhood
is also written as Nr (a). A deleted neighborhood of a is
4
4.1
Chapter Four: Dierentiation
Lecture 1: Denitions and Examples
Denition 4.1. Suppose that f : I R and x0 I. Here, I is an open
interval, possibly semi-innite or even R itself. If N (x0 ) is a deleted neighborhood of x0 , then the dierence quotient of
6
6.1
Chapter Six: Numerical Innite Series
Lecture 1: Basic Denitions and Results
Denition 6.1. Suppose that cfw_ak R is a sequence of real numbers. By
k=1
the innite series k=1 ak (or simply
ak ) is meant the sequence of partial
sums:
n
sn =
ak .
k=1
T
5
Chapter Five: Riemann Integration
5.1
Lecture 1: The Riemann-Darboux Integral
Denition 5.1. i A partition P of the interval [a, b] R is a nite set,
x0 := a < x1 < < xn1 < xn := b.
The mesh (or meshlength) of P is the maximum subinterval length, dened
by