Definition:Lower Bound of Sequence

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This page is about lower bounds of sequences which are bounded below. For other uses, see Definition: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 $\left({T, \preceq}\right)$ be an ordered set.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $T$.


Let $\left \langle {x_n} \right \rangle$ be bounded below in $T$ by $L \in T$.


Then $L$ is a lower bound of $\left \langle {x_n} \right \rangle$.


Real Sequence

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


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 $\sequence {x_n}$'.


Also see