nonincreasing

Nonincreasing, also called weakly decreasing, describes a sequence or function that does not increase as its input progresses. For a sequence (a_n), it is nonincreasing if a_{n+1} ≤ a_n for every n. Equivalently, a_1 ≥ a_2 ≥ a_3 ≥ ... . For a real-valued function f defined on an interval, f is nonincreasing if x ≤ y implies f(x) ≥ f(y).

Examples illustrate the idea. The sequence 5, 4, 4, 3, 1 is nonincreasing because each term is at most the previous one. The function f(x) = -x on [0, ∞) is nonincreasing since larger inputs yield smaller outputs. By contrast, a strictly decreasing sequence satisfies a_{n+1} < a_n for all n, while nonincreasing allows equality.

Relationship to related concepts. Nonincreasing is the opposite of nondecreasing; a nonincreasing sequence cannot rise as n increases. In many texts, nonincreasing is also described as monotone decreasing or monotone nonincreasing, though “monotone” alone can refer to either nonincreasing or nondecreasing. The concepts extend to functions: a nonincreasing function preserves the order in the opposite direction of its input.

Properties and implications. Any subsequence of a nonincreasing sequence is nonincreasing. If a nonincreasing sequence is bounded below, it converges to its infimum; without a lower bound it can diverge to minus infinity. In analysis, monotone (including nonincreasing) sequences and functions often enjoy convergence properties under mild boundedness assumptions, and they play a key role in various theorems and algorithms, such as those involving monotone sequences, orderings, and optimization.