Nichtincreasing

Nichtincreasing, also known as monotonically decreasing, is a property of a sequence or function. A sequence is called nichtincreasing if each term is less than or equal to the previous term. Formally, for a sequence denoted by $a_1, a_2, a_3, \ldots$, it is nichtincreasing if $a_n \geq a_{n+1}$ for all natural numbers $n$. Similarly, a function $f(x)$ is nichtincreasing over an interval if for any two points $x_1$ and $x_2$ in the interval, where $x_1 \leq x_2$, it follows that $f(x_1) \geq f(x_2)$.

This property is fundamental in various areas of mathematics, including calculus, discrete mathematics, and computer science. For instance, in calculus, the derivative of a nichtincreasing function is always less than or equal to zero over the interval where the function is nichtincreasing. In algorithms, analyzing the behavior of a nichtincreasing sequence can be crucial for understanding convergence or termination conditions.

Examples of nichtincreasing sequences include $5, 4, 4, 3, 1, 0, 0, \ldots$ or a sequence where every term is the same, such as $2, 2, 2, 2, \ldots$. A typical example of a nichtincreasing function is $f(x) = -x^2$ for $x \geq 0$, as its values decrease or stay the same as $x$ increases. The term "nichtincreasing" is often used in contrast to "strictly decreasing," where each term must be strictly less than the previous one.