Page 3 - Real Analysis I-e content (1)_Neat

Theorem:

b
Given real numbers a & b such that  ba      0 . Then a 

Proof:

a   b

If b  then in inequality 
a
2
 a  b  
a   b b  a
 b  
2 2
a b

Hence proved.

Intervals:

b
The set of all points between a& is called an interval. It is important to note intervals which
include their end points (or) not.

x : x satifies P the set of all real numbers x which satisfies property P.

Definition:

  a 
a,
b
a,
Assume a  the open interval  b  is defined to be the set  b  x : x   b
The closed interval   b is the set    a  x   b .
b 
a,
x :
a,

Real line R is referred as the open interval  ,   . A single point is considered as a
“degenerate” closed interval.
Integers:

Integers is a special subset of R

Definition:

A set of real numbers is called an inductive set if it has the following two properties.

(i) The number 1 is in the set.

(ii) For every x in the set R is also a inductive set.
Definition:

The real number is called a positive integer if it belong to every inductive set. The set of positive

integers is denoted by Z