Definition:Increasing/Sequence/Real Sequence

Definition

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

Then $\sequence {x_n}$ is increasing if and only if:

$\forall n \in \N: x_n \le x_{n + 1}$

Also known as

An increasing sequence is also known as an ascending sequence.

Some sources refer to an increasing sequence which is not strictly increasing as non-decreasing or monotone non-decreasing.

Some sources refer to an increasing sequence as a monotonic increasing sequence to distinguish it from a strictly increasing sequence.

That is, such that monotonic is being used to mean an increasing sequence in which consecutive terms may be equal.

$\mathsf{Pr} \infty \mathsf{fWiki}$ does not endorse this viewpoint.

Examples

Example: $\sequence 1$

The first few terms of the real sequence:

$S = \sequence 1_{n \mathop \ge 1}$

are:

$1, 1, 1, 1, \dotsc$

$S$ is both increasing and decreasing.

Also see

• Definition:Strictly Increasing Real Sequence
• Definition:Decreasing Real Sequence
• Definition:Monotone Real Sequence

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
• 1962: Bert Mendelson: Introduction to Topology ... (previous) ... (next): $\S 2.5$: Limits: Exercise $4$
• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.15$: Sequences: Definition $15.3$
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $\S 1.2$: Real Sequences: Definition $1.2.5$
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.15$: Monotone Sequences