Definition:Bounded Sequence
This page is about Bounded in the context of Sequence. For other uses, see Bounded.
Definition
A special case of a bounded mapping is a bounded sequence, where the domain of the mapping is $\N$.
Let $\struct {T, \preceq}$ be an ordered set.
Let $\sequence {x_n}$ be a sequence in $T$.
Then $\sequence {x_n}$ is bounded if and only if $\exists m, M \in T$ such that $\forall i \in \N$:
- $(1): \quad m \preceq x_i$
- $(2): \quad x_i \preceq M$
That is, if and only if it is bounded above and bounded below.
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 if and only if $\exists m, M \in \R$ such that $\forall i \in \N$:
- $m \le x_i$
- $x_i \le M$
Complex Sequence
Let $\sequence {z_n}$ be a complex sequence.
Then $\sequence {z_n}$ is bounded if and only if:
- $\exists M \in \R$ such that $\forall i \in \N: \cmod {z_i} \le M$
where $\cmod {z_i}$ denotes the complex modulus of $z_i$.
Normed Division Ring
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\sequence {x_n}$ be a sequence in $R$.
Then $\sequence {x_n}$ is bounded if and only if:
- $\exists K \in \R$ such that $\forall n \in \N: \norm {x_n} \le K$
Normed Vector Space
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space.
Let $\sequence {x_n}$ be a sequence in $X$.
Then $\sequence {x_n}$ is bounded if and only if:
- $\exists K \in \R$ such that $\forall n \in \N: \norm {x_n} \le K$
Metric Space
Let $M$ be a metric space.
Let $\sequence {x_n}$ be a sequence in $M$.
Then $\sequence {x_n}$ is a bounded sequence if and only if $\sequence {x_n}$ is bounded in $M$.
That is:
- $\exists K \in \R: \forall n, m \in \N: \map d {x_n, x_m} \le K$
Unbounded Sequence
A sequence which is not bounded is unbounded.
Also see
- Results about bounded sequences can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): bound: 2. (of a sequence)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): bound: 2. (of a sequence)
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): bounded sequence