Definition:Bounded Below Sequence
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This page is about Bounded Below in the context of Sequence. For other uses, see Bounded Below.
Definition
Let $\struct {T, \preceq}$ be an ordered set.
Let $\sequence {x_n}$ be a sequence in $T$.
Then $\sequence {x_n}$ is bounded below if and only if:
- $\exists m \in T: \forall i \in \N: m \preceq x_i$
Real Sequence
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 below if and only if:
- $\exists m \in \R: \forall i \in \N: m \le x_i$
Unbounded Below
$\sequence {x_n}$ is unbounded below if and only if there exists no $m$ in $T$ such that:
- $\forall i \in \N: m \preceq x_i$
Also see
- Definition:Bounded Below Mapping, of which a bounded below sequence is the special case where the domain of the mapping is $\N$.