Definition:Strictly Increasing/Sequence/Real Sequence
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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