Bounded Sequence

Bounded Below

A sequence is bounded below if all its terms are greater than or equal to a number, K, which is called the lower bound of the sequence.

an ≥ k

The greatest lower bound is called the infimum.

Bounded Above

A sequence is bounded above if all its terms are less than or equal to a number K', which is called the upper bound of the sequence.

an ≤ k'

The smallest upper bound is called the supremum.

Bounded Sequence

A sequence is bounded if it is bounded above and below, that is to say, if there is a number, k, less than or equal to all the terms of sequence and another number, K', greater than or equal to all the terms of the sequence. Therefore, all the terms in the sequence are between k and K'.

k ≤ an ≤ K'

Examples

Study the following sequences and determine if they are bounded.

1

3, 4/3, 1, 6/7,...

As the sequence is decreasing, 3 is an upper bound and the supremum.

a1000= 0.5012506253127.

a1000 000 = 0.5000012500006.

The limit is 0.5.

0.5 is a lower bound and the infimum.

Thus, the sequence is bounded.

1/2 < an 3

2

As the sequence is increasing, 1/2 is a lower bound and the infimum.

a1000= 0.999000999001.

a1000 000 = 0.999999000001.

The limit is 1.

1 is an upper bound and the supremum.

Thus, the sequence is bounded.

0.5 ≤ an < 1