Lecture 1
: Real Numbers, Functions [ Section 1.1 : Real Numbers ]
(ii)
Similarly, we say is bounded below if there exists t such that tr for all r.
(iii)
A set is said to be bounded if it is both bounded above and below, i.e., there exist s, t such that tr s for every r.
1.1.4
Example :
is bounded below by 1. In fact, every is bounded below. Archimedian property says that is not bounded above.
1.1.5
, the set of Integers :
For every , let be the unique element of such that. Let : = { ..., -2, -1, 0, 1, 2, ... }
Elements of are called integers. Clearly, is neither bounded above nor bounded below.
1.1.6
, the set of rational numbers :
For every , let be such that . The element is also denoted by . Let : = The setis called the set of rational numbers and the elements of the set \ are called the irrational numbers. Both, the rational and the irrational numbers have the following denseness property :
1.1.7
Denseness of rational and the irrational numbers:
For every real numbers x and y, with,there exist a rational and an irrational such that and .
, the set of Integers :
be the unique element of 
. Let 
: = { ..., -2, -1, 0, 1, 2, ... } 
, let 
be such that 
. The element 
is also denoted by 
. Let 
: = 
,there exist a rational
and an irrational
and
.
