Page 4 - Real Analysis I-e content (1)_Neat

Upper bounds, maximum element, Least upper bound

Definition:

Let S be a set of real numbers. If there is a real number b such that x  b x in S , then b is called

an upper bound for S and we say that S is bounded above by b .

If an upper bound b is also a member of S , the b is called the largest member (or) the
maximum element of S


b max S

Similarly for lower bound, bounded below we have b m in S
Definition:

Let S be a set of real numbers bounded above.

A real number b is called a least upper bound for S if it has the following two properties.

(i) b is an upper bound for S .
(ii) No number less than b is an upper bound for S .

Example:
   1 , 0 , max element is 1 is least upper bound.
S

S    1 , 0 1 is least upper bound for S , but S has no maximum element.

Definition:

The least upper bound is denoted by

b  sup S

If S has maximum element then

max S  sup S

Definition:

The greatest lower bound is denoted by

b inf S

If S has minimum element then

min S  inf S