# Definition:Strictly Decreasing/Sequence

## Definition

Let $\left({S, \preceq}\right)$ be a totally ordered set.

Then a sequence $\left \langle {a_k} \right \rangle_{k \in A}$ of terms of $S$ is strictly decreasing iff:

$\forall j, k \in A: j < k \implies a_k \prec a_j$

### 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 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.

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