Material for PMA104
Ivan Todorov
March 5, 2008
The file contains the material covered in PMA104. The statements are
proved in class unless otherwise stated.
1
Preliminaries
Sets, operations with sets, properties of the operations with sets. The sets
N
,
Z
,
Q
,
R
,
C
.
Notation for intervals: [
a, b
] for closed intervals, (
a, b
) for
open intervals etc.
Mappings between sets. Injective, surjective and bijective mappings.
Sequences of real numbers.
Convergence of sequences.
Arithmetics of
limits: lim
n
→∞
(
a
n
+
b
n
) = lim
n
→∞
a
n
+ lim
n
→∞
b
n
etc. The Sandwich Rule.
Basic inequalities and identities: the triangle inequality, Newton’s Bino
mial Formula,
a
n

b
n
= (
a

b
)(
a
n

1
+
a
n

2
b
+
· · ·
+
b
n

1
).
2
Analysis
2.1
Subsequences
Definition of a subsequence.
Notation: (
a
n
k
)
∞
k
=1
.
Every subsequence of a
convergent sequence converges to the same limit. Examples: (
1
2
n
), (
1
n
2
).
Definition of a function
f
:
I
→
R
.
The domain is always an interval.
The definition of the graph of a function. Review of basic functions and how
1
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their graphs look like: the constant and the identity functions,
x
n
,
n
√
x
,
a
x
(
a >
1), log
x
, sin
x
, cos
x
, tan
x
, cot
x
.
The step function
f
: (

1
,
1)
→
R
given by
f
(
x
) =
(

1
if

1
< x <
0
1
if 0
≤
x <
1
.
The Dirichlet function
χ
:
R
→
R
given by
χ
(
x
) =
(
1
if
x
∈
Q
0
if
x
6∈
Q
.
Definition 2.1
Let
I
be an open interval,
f
:
I
→
R
be a function, and
x
0
3
I
.
We say that
f
is continuous at
x
0
if for every
² >
0
there exists
δ >
0
such that if
x
∈
I
,

x

x
0

< δ
then

f
(
x
)

f
(
x
0
)

< ²
. If a function
f
is not continuous at
x
0
we say that
f
is discontinuous at
x
0
.
Proposition 2.2
Let
I
be an open interval,
f
:
I
→
R
be a function, and
x
0
3
I
. The following are equivalent:
(i)
f
is continuous at
x
0
;
(ii) for every sequence
(
x
n
)
∞
n
=1
⊆
I
with
lim
n
→∞
x
n
=
x
0
we have that
lim
n
→∞
f
(
x
n
) =
f
(
x
0
)
.
Examples (i)
The constant function is continuous at every point.
(ii)
The identity function is continuous at every point.
(iii)
The function
f
:
R
→
R
given by
f
(
x
) =
λx
,
x
∈
R
, where
λ
is a fixed
real number, is continuous at every point.
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 Spring '08
 TOD
 Linear Algebra, Vector Space, Sets, Linear combination

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