Definition:Strictly Increasing/Sequence/Real Sequence

Definition

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

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

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

Also known as

A strictly increasing sequence is also referred to as ascending or strictly ascending.

Some sources refer to a strictly increasing sequence as an increasing sequence, and refer to an increasing sequence which is not strictly increasing 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: $\sequence {2^n}$

The first few terms of the real sequence:

$S = \sequence {2^n}_{n \mathop \ge 1}$

are:

$2, 4, 8, 16, \dotsc$

$S$ is strictly increasing.

Also see

• Definition:Increasing Real Sequence
• Definition:Strictly Decreasing Real Sequence
• Definition:Strictly 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
• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.15$: Sequences
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.15$: Monotone Sequences