Definition:Bounded Sequence/Real

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

Definition

Let $\sequence {x_n}$ be a real sequence.

Then $\sequence {x_n}$ is bounded if and only if $\exists m, M \in \R$ such that $\forall i \in \N$:

$m \le x_i$

$x_i \le M$

That is, if and only if it is bounded above and bounded below.

Unbounded

$\sequence {x_n}$ is unbounded if and only if it is not bounded.

Examples

Example: $1 - \frac 1 n$

Let $\sequence {s_n}_{n \mathop \ge 1}$ be the real sequence defined as:

$s_n = 1 - \dfrac 1 n$

That is:

$\sequence {s_n}$ is the sequence of $1$ minus the reciprocals of the strictly positive integers.

Then $\sequence {s_n}$ is bounded.

Also see

• Results about bounded 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 $4$
• 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