Nondecreasingnonincreasing

Nondecreasingnonincreasing is an informal way to describe a sequence, function, or other ordered data that satisfies both the nondecreasing and nonincreasing conditions. A sequence (a_n) is nondecreasing if a_n ≤ a_{n+1} for all n, and nonincreasing if a_n ≥ a_{n+1} for all n.

If a sequence is both nondecreasing and nonincreasing, then for every n we have a_n ≤ a_{n+1} and a_n ≥ a_{n+1}. This forces a_n = a_{n+1} for all n, so all terms are equal and the sequence is constant.

The same conclusion holds for real-valued functions defined on an interval. If f is nondecreasing and nonincreasing on that interval, then for any x < y we have f(x) ≤ f(y) and f(x) ≥ f(y), which implies f(x) = f(y) for all x and y. Therefore, such a function is constant on the interval.

Examples help illustrate the idea. The constant sequence 5, 5, 5, 5 is both nondecreasing and nonincreasing. By contrast, the sequence 1, 2, 2, 3 is nondecreasing but not nonincreasing, while 4, 4, 3, 2 is nonincreasing but not nondecreasing.

In analysis and related fields, the property of being both nondecreasing and nonincreasing is used to identify constant behavior. When a function or sequence possesses both monotone properties, its values do not vary across its domain or index set. Related concepts include monotone (in general) and constant functions.