Introduction to Real
Analysis
Dr. Weihu Hong
Clayton State University
9/9/2008
Properties on the absolute
value



|x| = x if x>0; -x if x≤0.
The geometric meaning of |x|: the distance between
x and 0.
Theorem 2.1.2 (Properties on |x|)
|-x| = |x| for all x єR
 |xy| =|x||y| for all x, y єR
 |x| =
x 2 for all xєR
 If r > 0, then |x| < r if and only if –r < x < r.
 -|x| ≤ x ≤ |x| for all xєR
You should know how to prove each of these statements!

Properties on the absolute
value



Theorem 2.1.3 (Triangle inequality)
For all x, yєR, we have |x + y| ≤ |x| + |y|.
Proof: Consider 0 ≤ (x + y)²
Corollary 2.1.4
For all x, y, zєR, we have
|x – y|≤ |x – z| + |z – y|
 ||x| - |y|| ≤ |x – y|
Make sure you know how to prove these statements!

Neighborhood of a point

Definition 2.1.6 (ε-neighborhood of the point p)
Let pєR and let ε > 0. The set
N  {x  R :| x  p |  }
is called an ε-neighborhood of the point p.
Note: it is the same if we write
N   {x  R : p    x  p   }
Convergence of a Sequence

Definition 2.1.7

{
p
}
A sequence n n 1 in R is said to converge if there
exists pєR such that for every ε>0, there exists a
positive integer K such that pn  N ( p) for all n≥K. In
this case, we say that { pn }n 1 converges to p, or that p
is the limit of the sequence { pn }n 1 , and we write
lim pn  p or pn  p
n 
If { pn }n 1 does not converge, then { pn }n 1 is said to
diverge.
Bounded Sequence




{
p
}
A sequence
n n 1 in R is said to be bounded if
there exists a positive constant M such that
| pn | M
for all nєN.
How would you define a sequence is unbounded?
Theorem 2.1.10


(a) If a sequence in R converges, then its limit is unique.
(b) Every convergent sequence in R is bounded.
You need to know how to prove both of these!
More theorems on limit

Theorem 2.2.1


{
a
}
and
{
b
}
n n 1 are convergent sequences of
If n n 1
an  a and lim bn  b
real numbers with lim
n 
n 
then

lim (an  bn )  a  b
n 
lim (anbn )  a b
n 
Furthermore, if a  0
and an  0 for all n, then
bn b

n  a
a
n
lim
More theorems on limit

Corollary 2.2.2
If {an }n 1 is a convergent sequence of real
an  a
numbers with lim
n 
then for any cєR,

lim (an  c)  a  c
n 
lim (can )  ca
n 
More theorems on limit

Theorem 2.2.3
Let {an }n 1 and {bn }n 1 be sequences of real numbers
with
lim an  0 and {bn }is bounded
n 
then

lim (anbn )  0
n 
More theorems on limit

Theorem 2.2.4 (Squeeze theorem)
Let {an }n 1 ，{bn }n 1 , and {cn }n 1 be sequences of real
numbers for which there exists KєN such that
an  bn  cn for all n  N , n  K ,
and that lim an  lim cn  L
n 
then

lim bn  L
n 
n 