Definition:Bounded Below Sequence/Real

This page is about Bounded Below Real Sequence. For other uses, see 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

$\sequence {x_n}$ is unbounded below if and only if there exists no $m$ in $\R$ such that:

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

Examples

Strictly Positive Integers

Let $\sequence {s_n}$ be the real sequence defined as:

$s_n = n$

That is:

$\sequence {s_n}$ is the sequence of strictly positive integers.

Then $\sequence {s_n}$ is bounded below.

Also see

• Definition:Bounded Above Real Sequence
• Definition:Bounded Real Sequence

• 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 $\N$.

• Results about bounded below real sequences can be found here.

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 5$: Limits: Exercise $4$
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.2$: Sequences
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): bounded sequence
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): bounded sequence