This document defines upper and lower bounds of sets of real numbers. An upper bound is a number that is greater than or equal to all elements of the set, while a lower bound is less than or equal to all elements. A set is bounded if it has both an upper and lower bound. The supremum is the least upper bound of a set, while the infimum is the greatest lower bound. An example finds the supremum and infimum of the set S = {x: 3x^2 - 10x + 3 <0}, which is determined to be bounded between 1/3 and 3, so the supremum is 3 and the infimum is 1/3.

1. Boundedness of
– { SETS } –
Course Title : Real Analysis-1
Course Code
MTH-211

2. Upper bound & Lower bound
Upper bound of a set: Let S be any subset of the set R of real
numbers. If there exists a real number u such that x ≤u, x  S, then u is
called an upper bound of the set S.
Lower bound of a set : Let S be any subset of the set R of real numbers.
If there exists a real number l such that Is l ≤ x, x S, then l is called a
lower bound of the set S.
Example: Let S = {…-3, -2, -1} Here -1 is an upper bound of S.
Example : Let S = {1,2,3,4….} Here lower bound of S = 1

3. Bounded & Unbounded sets
Bounded set : Let S be any subset of the set R of real numbers. If there exists two real numbers u
and I such that,
l  x  u
then the set S is called bounded.
Example : Let S = {1, 3, 5, 7}
Here 1 is a lower bound and 7 is an upper bound
Unbounded set: Let S is a set. If there exist no real number I and u such that. I≤x≤ u
hold, x  S, then S is called an unbounded set.
Example : The set R of real numbers is an unbounded set.
because it is nether bounded above nor bounded below.

4. Supremum & Infimum
Supremum :The least of all the upper bounds of a set is called its supremum
or the least upper bound.
Example: Let S = {1, 2. 3. 4} Here 4 is an upper bound of S
Supremum of S = 4.
supS=4]
Infimum : The greatest of all the lower bounds of a set is called its infimum or
the greatest lower bound.
Example: Let S = (1, 2, 3, 4) Here ,1 is a lower bound of S..
Infimum of S = 1.
infS=1]

5. Find the supremum and infimum of the set S = {x: 3x² - 10x + 3 <0}
Solution : Given S= { x : 3x² - 10x + 3 < 0}
Now 3x2-10x+3<0
⇒ 3x²-9x-x+3<0
⇒3x(x-3) -1(x-3) <0
⇒(3x - 1) (x-3) <0
This is true when
(i) 3x-1>0 and x-3<0
⇒ x> 1/3 and x<3
⇒ 1/3 < x < 3
Or,
(ii) 3x-1 <0 and x-3>0
⇒x< and x>3
When x <1/3 then 3x-1 <0 and x-3<0
When x>3 then 3x-1>0 and x-3>0
Thus, for x < 1/3 or x > 3
we have (3x - 1)(x-3) > 0
S=(x : 3x² - 10x +3<0)
⇒ S = {x: 1/3 < x < 3}
sup S = 3
inf S = 1/3 (Ans)

6. Thanks
All