Definition:Upper Bound of Sequence

This page is about Upper Bound in the context of Bounded Above Sequence. For other uses, see 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 $\struct {T, \preceq}$ be an ordered set.

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

Let $\sequence {x_n}$ be bounded above in $T$ by $H \in T$.

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

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