# Non-increasing sequence

In mathematics a non-increasing sequence is a type sequence, an ordered list of objects, with an additional property on the values of its items. The property of being non-increasing is a local property in that if all adjacent items in the sequence have the property then the sequence over all has the property.

The property of being a decreasing sequence is a stronger requirement since any "decreasing sequence" is a "non-increasing sequence", but the converse is not true. A special example of a non-increasing sequence is the constant sequence where all items in the sequence have identical value.

Definition

Let "X" be a set, and "⊆" a partial order on "X". Let α be a sequence of ordinals. We say that a sequence of elements "x"α of "X" is non-increasing if for each ordinal α and each ordinal β>α, we have "x"β⊆"x"α.

Algebraically speaking, a sequence $\left(a_n\right)_\left\{n=1\right\}^\left\{infty\right\}$ is non-increasing if $forall ninmathbb N:a_ngeq a_\left\{n+1\right\}$.

*Monotonic function
*decreasing sequence

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