# Definition:Bounded Below Sequence

Jump to navigation
Jump to search

*This page is about Bounded Below in the context of Sequence. For other uses, see Bounded Below.*

## Definition

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

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

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

- $\exists m \in T: \forall i \in \N: m \preceq x_i$

### 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 below** if and only if:

- $\exists m \in \R: \forall i \in \N: m \le x_i$

## Unbounded Below

$\sequence {x_n}$ is **unbounded below** if and only if there exists no $m$ in $T$ such that:

- $\forall i \in \N: m \preceq x_i$

## Also see

- Definition:Bounded Below Mapping, of which a
**bounded below sequence**is the special case where the domain of the mapping is $\N$.