# 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.

**a _{n} ≥ 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.

**a _{n }≤ 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 ≤ a _{n} ≤ 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.

a_{1000}= 0.5012506253127.

a_{1000 000 }= 0.5000012500006.

The limit is 0.5.

0.5 is a lower bound and the infimum.

Thus, the sequence is bounded.

1/2 < a_{n} ≤ 3

2

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

a_{1000}= 0.999000999001.

a_{1000 000 }= 0.999999000001.

The limit is 1.

1 is an upper bound and the supremum.

Thus, the sequence is bounded.

0.5 ≤ a_{n }< 1