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-

  1. Upper Bound
  2. 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..