Definition:Lower Bound of Sequence

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

Definition

A special case of a lower bound of a mapping is a lower bound of a 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 $T$.

Let $\sequence {x_n}$ be bounded below in $T$ by $L \in T$.

Then $L$ is a lower bound of $\sequence {x_n}$.

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.

Let $\sequence {x_n}$ be bounded below in $T$ by $L \in \R$.

Then $L$ is a lower bound of $\sequence {x_n}$.