Definition:Strictly Decreasing/Sequence/Real Sequence
Definition
Let $\sequence {x_n}$ be a sequence in $\R$.
Then $\sequence {x_n}$ is strictly decreasing if and only if:
- $\forall n \in \N: x_{n + 1} < x_n$
Also known as
A strictly decreasing sequence is also referred to as strictly order-reversing.
Some sources use the term descending sequence or strictly descending sequence.
Some sources refer to a strictly decreasing sequence as a decreasing sequence, and refer to a decreasing sequence which is not strictly decreasing as a monotonic decreasing sequence to distinguish it from a strictly decreasing sequence.
That is, such that monotonic is being used to mean a decreasing sequence in which consecutive terms may be equal.
$\mathsf{Pr} \infty \mathsf{fWiki}$ does not endorse this viewpoint.
Examples
Example: $\sequence {n^{-1} }$
The first few terms of the real sequence:
- $S = \sequence {n^{-1} }_{n \mathop \ge 1}$
are:
- $1, \dfrac 1 2, \dfrac 1 3, \dfrac 1 4, \dotsc$
$S$ is strictly decreasing.
Also see
- Results about decreasing 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
- 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
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): decreasing sequence