# Definition:Bounded Below Sequence/Real

< Definition:Bounded Below Sequence(Redirected from Definition:Bounded Below Real Sequence)

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*This page is about real sequences which are bounded below. For other uses, see Definition:Bounded Below.*

## Definition

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

$\left \langle {x_n} \right \rangle$ is **unbounded below** if and only if there exists no $m$ in $\R$ such that:

- $\forall i \in \N: m \le x_i$

## Also see

- Definition:Bounded Below Real-Valued Function, of which a
**bounded below real sequence**is the special case where the domain of the real-valued function is $\N$.

## Sources

- 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 2.5$: Limits: Exercise $4$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.2$: Sequences