Bounded Sequence: Definition, Examples

A sequence {x_n} is called a bounded sequence if k ≤ x_n ≤ K for all natural numbers n. For example, {1/n} is a bounded sequence since 0 < 1/n ≤ 1 for all n.

Let us now learn about bounded sequence.

Bounded Sequence Definition

A sequence {x_n} is said to be bounded if

k ≤ x_n ≤ K for all n ∈ ℕ

for some real numbers k and K.

Note: Here the numbers k and K are respectively called a lower bound and an upper bound of the sequence.

REMARK:

If |xn| ≤ K for all n ∈ ℕ, for some positive real number K, then the sequence {xn} is bounded.

Bounded Sequence Examples

Ex1: The sequence {1/n} is bounded.

This is because 0 < 1/n ≤ 1 for all n ∈ ℕ.

Here 0 and 1 are the lower and upper bounds respectively.

Ex2:

The sequence $\left\{\dfrac{n}{n+1} \right\}$ is bounded. The reason is the following:

Here $x_n=\dfrac{n}{n+1}$.

Now, $|x_n|=|\dfrac{n}{n+1}| = \dfrac{n}{n+1} <1$ for all n ∈ ℕ.

As |x_n| ≤ 1 for all n ∈ ℕ, the given sequence {n/n+1} is bounded.

Note that -1 and 1 are the lower and upper bounds of the sequence {n/n+1} respectively.

Ex3:

The sequence $\left\{\dfrac{(-1)^n n^2}{n^2+1} \right\}$ is bounded. Here is the reason:

We have $x_n=\dfrac{(-1)^n n^2}{n^2+1}$.

Now,

$|x_n|=|\dfrac{(-1)^n n^2}{n^2+1}| = \dfrac{n^2}{n^2+1} <1$ for all n ∈ ℕ.

As $|\dfrac{(-1)^n n^2}{n^2+1}|$ ≤ 1 for all n ∈ ℕ, the given sequence {(-1)ⁿn²/(n²+1)} is bounded.

Note that -1 and 1 are respectively the lower and upper bounds of the sequence {(-1)ⁿn²/(n²+1)}.

Ex4:

The sequence {n} is not bounded.

Note 1 is the lower bound of the sequence {n} and it has no upper bounds. Thus, the sequence {n} is unbounded.

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