# Bounded Above & Bounded Below Sequences

This post talks about the Bounded Sequence & the bounds of sequences. – the Bounded above & the Bounded below sequence. If a sequence has bounds then there arises two cases- that are either upper bound or lower bound.

Let us discuss about them in details..

# Bounded Sequence:

If a sequence has some value which binds or restrict further expansion of the sequence, then-

that value is the * Bound *of that sequence & the corresponding sequence is the

*.*

**Bounded Sequence**Normally, a sequence has two bounds-

- Upper Bound
- Lower Bound

# Upper Bound of a Sequence (lub):

A sequence has an upper bound if it has some higher value which restrict the further expansion of sequence (i.e. limit) in upper direction, is a bounded above sequence.

If a value among some upper values for any particular sequence is selected as the * least upper bound (lub)* , then that sequence always converges to that value(lub).

Supremum is another name for the Least Upper bound, i.e.(lub).

A sequence having supremum is the * bounded above Sequence*.

A bounded above sequence always converges to its upper bound.

#### Example:

If we define a sequence of even numbers upto 10, then it is bounded above at 10. i.e. it has a supremum (lub) 10.

S={2,4,6,8,10}

# Lower Bound of a Sequence (glb):

A sequence has an lower bound if it has some lower value which restrict the further expansion of sequence in below direction, is a bounded below sequence.

If a value among some lower values for any particular sequence is selected as the * greatest lower bound (glb)* , then that sequence always converges to that value(glb).

Infimum is another name for the Greatest Lower bound, i.e.(glb).

A sequence having infimum is the * bounded below Sequence*.

A bounded below sequence always converges to its lower bound.

#### Example:

If we define a sequence of even numbers upto 10, then it is bounded below by 2. i.e. it has a infimum (glb) 2.

S={2,4,6,8,10}

It’s about the bounds of the sequences. We’ll discuss more topics further..