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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 $(a_n)_{n=1}^{infty}$ is non-increasing if $forall ninmathbb N:a_ngeq a_{n+1}$ .

Non-increasing sequence

Definition

See also