Neverdecreasing

Neverdecreasing is an informal term used to describe a sequence, function, or process that never decreases in value. In mathematical terminology it is equivalent to being nondecreasing or monotone nondecreasing. The idea is that the value never goes down as one progresses along the sequence or across the domain of the function, though it may stay the same for some steps.

For a sequence (a_n), neverdecreasing means a_{n+1} ≥ a_n for all n. For a real-valued function f defined on an interval, it means that x ≤ y implies f(x) ≤ f(y). The term allows equality, so a flat step is permitted. It is distinct from strictly increasing, where the inequality is always strict (a_{n+1} > a_n or f(x) < f(y) for x < y).

Examples illustrate the idea: the sequence 1, 2, 2, 5, 9 is never decreasing; the function f(x) = x^2 is nondecreasing on [0, ∞). In contrast, f(x) = |x| is not nondecreasing on the whole real line, though it is on certain subintervals.

Properties and implications: a nondecreasing sequence that is bounded above converges to its supremum (monotone convergence). If unbounded above, such a sequence diverges to infinity. Nondecreasing functions on an interval are of bounded variation and have well-behaved limits from the right or left at internal points.

Applications appear in analysis, probability (cumulative distribution functions are nondecreasing), economics, and computer science wherever an order-preserving growth pattern is relevant. See also monotone, nondecreasing, and monotonic function for related concepts.