Definition:Strictly Decreasing/Sequence
Definition
Let $\struct {S, \preceq}$ be a totally ordered set.
Then a sequence $\sequence {a_k}_{k \mathop \in A}$ of terms of $S$ is strictly decreasing if and only if:
- $\forall j, k \in A: j < k \implies a_k \prec a_j$
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 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.
Also see
- Definition:Decreasing Sequence
- Definition:Strictly Increasing Sequence
- Definition:Strictly Monotone Sequence
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): decreasing sequence
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): decreasing sequence