Nonincrease

Nonincrease, commonly written nonincreasing, is a term used in mathematics to describe a quantity that does not increase as its input grows. The word can function as both a noun (the property of nonincreasing) and an adjective (a nonincreasing sequence or function).

In sequences, a sequence (a_n) is nonincreasing if a_{n+1} ≤ a_n for every index n. Such sequences may stay constant over steps but never rise. If a nonincreasing sequence is bounded below, it converges to its infimum; without a lower bound it may diverge to negative infinity. For example, the sequence 5, 5, 3, 3, 2, 1, 1 is nonincreasing.

In the context of functions, a function f defined on an ordered set is nonincreasing if x ≤ y implies f(x) ≥ f(y). Examples include f(x) = -x on the real numbers, and f(x) = 1/x on the positive real axis. These examples illustrate how the value of the function does not increase as the input grows.

Nonincrease is a specific instance of monotonicity and is important in analysis, optimization, and algorithm design. Monotone (nonincreasing) behavior often underpins convergence proofs, ordering arguments, and stability analyses. Properties of nonincreasing functions on intervals include predictable direction of change and, in many contexts, a controlled structure of discontinuities, such as jump discontinuities in the case of real-valued monotone functions.