Definition: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 decreasing if and only if:
- $\forall j, k \in A: j < k \implies a_k \preceq a_j$
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 decreasing if and only if:
- $\forall n \in \N: x_{n + 1} \le x_n$
Also known as
A decreasing sequence is also referred to as order-reversing.
Some sources use the term descending sequence.
Some sources refer to a decreasing sequence which is not strictly decreasing as non-increasing or monotone non-increasing.
Some sources refer to a decreasing sequence 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
- Results about decreasing sequences can be found here.
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