Cardinalities of monotone natural-number functions
= Cardinalities of monotone natural-number functions
<Nondecreasing functions> $\mathbb N\to\mathbb N$ form an uncountable set: every subset $A$ is encoded by the strictly increasing function $n+\sum_{k=1}^{n}\mathbf1_A(k)$. In contrast, <nonincreasing functions> form a countable set because each is determined by a finite initial segment and its eventual constant value.
// Target: classical-mechanics.bigb