Definition:Bounded Above Sequence

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

Definition

Let $\struct {T, \preceq}$ be an ordered set.

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

Then $\sequence {x_n}$ is bounded above if and only if:

$\exists M \in T: \forall i \in \N: x_i \preceq M$

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 above if and only if:

$\exists M \in \R: \forall i \in \N: x_i \le M$

$\sequence {x_n}$ is unbounded above if and only if there exists no $M$ in $T$ such that:

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

Definition:Bounded Above Mapping, of which a bounded above sequence is the special case where the domain of the mapping is $\N$.