# Definition:Bounded Below Sequence

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:

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