Definition:Monotone (Order Theory)/Sequence

Definition

Let $\struct {S, \preceq}$ be a totally ordered set.

A sequence $\sequence {a_k}_{k \mathop \in A}$ of elements of $S$ is monotone if and only if it is either increasing or decreasing.

Real Sequence

The above definition for sequences is usually applied to real number sequences:

Let $\sequence {x_n}$ be a sequence in $\R$.

Then $\sequence {x_n}$ is monotone if and only if it is either increasing or decreasing.

Also known as

A monotone sequence can also be called a monotonic sequence.

Also see

• Definition:Strictly Monotone Sequence

• Results about monotone sequences can be found here.

Sources

• 1919: Horace Lamb: An Elementary Course of Infinitesimal Calculus (3rd ed.) ... (previous) ... (next): Chapter $\text I$. Continuity: $2$. Upper or Lower Limit of a Sequence (Footnote $*$)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): monotonic sequence
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): monotonic sequence
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): monotonic sequence