Definition:Decreasing/Sequence/Real Sequence

From ProofWiki
Jump to navigation Jump to search


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.


Example: $\sequence 1$

The first few terms of the real sequence:

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


$1, 1, 1, 1, \dotsc$

$S$ is both increasing and decreasing.

Also see

  • Results about decreasing sequences can be found here.