# Definition:Bounded Sequence

## Definition

A special case of a bounded mapping is a bounded sequence, where the domain of the mapping is $\N$.

Let $\struct {T, \preceq}$ be an ordered set.

Let $\sequence {x_n}$ be a sequence in $T$.

Then $\sequence {x_n}$ is bounded if and only if $\exists m, M \in T$ such that $\forall i \in \N$:

$(1): \quad m \preceq x_i$
$(2): \quad x_i \preceq M$

That is, if and only if it is bounded above and bounded below.

### Real Sequence

The concept is usually encountered where $\struct {T, \preceq}$ is the set of real numbers under the usual ordering: $\struct {\R, \le}$:

Let $\sequence {x_n}$ be a real sequence.

Then $\sequence {x_n}$ is bounded if and only if $\exists m, M \in \R$ such that $\forall i \in \N$:

$m \le x_i$
$x_i \le M$

### Complex Sequence

Let $\sequence {z_n}$ be a complex sequence.

Then $\sequence {z_n}$ is bounded if and only if:

$\exists M \in \R$ such that $\forall i \in \N: \cmod {z_i} \le M$

where $\cmod {z_i}$ denotes the complex modulus of $z_i$.

### Normed Division Ring

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.

Let $\sequence {x_n}$ be a sequence in $R$.

Then $\sequence {x_n}$ is bounded if and only if:

$\exists K \in \R$ such that $\forall n \in \N: \norm {x_n} \le K$

### Normed Vector Space

Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.

Let $\sequence {x_n}$ be a sequence in $X$.

Then $\sequence {x_n}$ is bounded if and only if:

$\exists K \in \R$ such that $\forall n \in \N: \norm {x_n} \le K$

### Metric Space

Let $M$ be a metric space.

Let $\sequence {x_n}$ be a sequence in $M$.

Then $\sequence {x_n}$ is a bounded sequence if and only if $\sequence {x_n}$ is bounded in $M$.

That is:

$\exists K \in \R: \forall n, m \in \N: \map d {x_n, x_m} \le K$

## Unbounded Sequence

A sequence which is not bounded is unbounded.