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Real Function Relation Theory Sequence Monotonically Nondecreasing Monotonically Increasing Monotonically Decreasing

1	Real Function
1	Sequence
1	Relation Theory
4	Monotonically Increasing
5	Monotonically Decreasing
5	Monotonically Nondecreasing

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Monotonically Nonincreasing

A sequence is monotonically nonincreasing if

$\displaystyle s_m \le s_n$ for all $\displaystyle m > n$

Similarly, a real function is monotonically nonincreasing if

$\displaystyle f(x) \le f(y)$ for all $\displaystyle x > y$

Compare this to monotonically decreasing.

Conflict note. In some contexts, such as [1], this is called monotonically decreasing (in turn, our “monotonically decreasing” is called “strictly decreasing”). This is unfortunately counter-intuitive, since a sequence or function that is “flat” (such as ) is somehow “decreasing.” Beware!

Examples

$(s_n) = 1, 0, -1, -2, \ldots$ is monotonically nonincreasing. It is also monotonically decreasing.
$(s_n) = 1, 1, 1, 1, \ldots$ is nonincreasing but not monotonically decreasing.
$(s_n) = (\frac{1}{n+1})$ is nonincreasing (note that is nonnegative).
$(s_n) = 1, 1, 2, 1, 1, \ldots$ is not nonincreasing. It also happens to fail to be monotonically nondecreasing.
$(s_n) = 1, 2, 3, 4, 5, \ldots$ is not nonincreasing, rather it is nondecreasing (and monotonically increasing).

Bibliography

1: ``monotonically decreasing,'' from the NIST Dictionary of Algorithms and Data Structures, Paul E. Black, ed.