Definition:Upper Bound of Sequence

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This page is about upper bounds of sequences which are bounded above. For other uses, see Definition:Upper Bound.

Definition

A special case of an upper bound of a mapping is an upper 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 above in $T$ by $H \in T$.


Then $H$ is an upper 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 above by $H \in \R$.


Then $H$ is an upper bound of $\sequence {x_n}$.


Also see