ECE 3250
MORE ON REAL AND COMPLEX NUMBERS
Fall 2008
My purpose in this handout is to summarize fairly briskly the main results on sequences
and series of real and complex numbers. I’ll be lifting some of the definitions (e.g. con
vergent sequences, Cauchy sequences, etc.) from the
NUMBERS
handout, but I think
it’s useful to collect the principal facts in one place in a kind of bulletedlist format. The
lack of a lot of intervening text makes for some highdensity mathematics, but the layout,
I hope, will facilitate easy reference.
To start with, I’ll assume you have some basic familiarity with the real numbers
R
and
the complex numbers
C
. I’ll assume you understand the algebra of complex numbers at
the level of the
ALGEBRA OF COMPLEX NUMBERS
handout. For a real number
a
,

a

denotes the absolute value of
a
; for a complex number
c
,

c

denotes the magnitude
of
c
. So if
c
=
a
+
jb
, with
a
and
b
in
R
, then

c

=
p
a
2
+
b
2
.
The distance between two real numbers
a
and
b
is the absolute value of
a

b
and the
distance between two complex numbers
c
1
and
c
2
is the magnitude of
c
1

c
2
. To avoid
having to type “real or complex numbers” a zillion times, I’ll use the notation
F
to denote
the phrase “
R
or
C
.” The “F” is supposed to mean “field.”
Notational Convention:
In what follows,
F
=
R
or
C
.
Definition 1:
A
sequence in
F
is an ordered list of elements of
F
indexed by
N
. We
use notation such as
{
a
n
}
or
{
c
n
}
to denote such a sequence. So, for example,
{
a
n
}
=
a
0
, a
1
, a
2
, a
3
,
. . . .
Definition 2:
We say that a sequence
{
a
n
}
in
F
converges
to ¯
a
∈
F
if and only if the
distance between
a
n
and ¯
a
approaches zero as
n
→ ∞
. In this case, we write
lim
n
→∞
a
n
= ¯
a .
A precise mathematical definition of convergence:
{
a
n
}
converges to ¯
a
if and only if for
every
>
0 there exists an integer
N >
0 such that

a
n

¯
a

<
for every
n > N
.
Fact 1:
A sequence
{
c
n
=
a
n
+
jb
n
}
in
C
converges to ¯
c
= ¯
a
+
j
¯
b
∈
C
if and only if
{
a
n
}
converges to ¯
a
and
{
b
n
}
converges to
¯
b
.
Proof:
First of all, for every
n
∈
N
,

c
n

¯
c

=
q

a
n

¯
a

2
+

b
n

¯
b

2
,
If
{
c
n
}
converges to ¯
c
, then for every
>
0 we can find
N >
0 so that

c
n

¯
c

<
when
n > N
. Hence for
n > N
, we have
q

a
n

¯
a

2
+

b
n

¯
b

2
<
,
which implies that both

a
n

¯
a

<
and

b
n

¯
b

<
for every
n > N
. Accordingly,
{
a
n
}
converges to ¯
a
and
{
b
n
}
converges to
¯
b
.
Conversely, if
{
a
n
}
converges to ¯
a
and
{
b
n
}
converges to
¯
b
, then for every
>
0 we
can find
N >
0 so that both

a
n

¯
a

<
2
/
2 and

b
n

¯
b

<
2
/
2 when
n > N
. Hence for
n > N
, we have

c
n

¯
c

<
p
2
/
2 +
2
/
2 =
.
1
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2
Accordingly,
{
c
n
}
converges to ¯
c
.
In words, Fact 1 states that a sequence of complex numbers converges if and only if
the realpart sequence and imaginarypart sequence both converge, in which case the limit
of the real parts is the real part of the limit, and the limit of the imaginary parts is the
imaginary part of the limit.
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 '04
 LIPSON
 lim, Order theory, upper bound, Limit of a sequence, Cauchy sequence

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