**Upper Bound and Bounded Above:-**

Let

**S**is a subset of

**R.**And a∈R. Then a is called upper bound of S if

**x**

**≤a**for all x∈S.

And then S is called bounded above.

**Lower Bound and Bounded Below:-**

Let

**S**is a subset of

**R.**And a∈R. Then a is called lower bound of S if

**a**

**≤x**for all x∈S.

And then S is called bounded below.

**Least Upper Bound:-**

Let

**S**is a subset of

**R.**And a∈R. Then a is called least upper bound of S if it is satisfied two axioms.

1)

**x**

**≤a**for all x∈S.

2) there are exist a Upper Bound

**t**such that

**a**

**≤t.**

I.e. lub is less or equal all other upper bounds.

**Greatest lower Bound:-**

Let

**S**is a subset of

**R.**And a∈R. Then a is called least upper bound of S if it is satisfied two axioms.

1) a

**≤x for**all x∈S.

2) there are exist a lower Bound w such that

**w**

**≤a.**

I.e. glb is greater or equal all other lower bounds.