Bounded nondecreasing integer sequences are eventually constant
= Bounded nondecreasing integer sequences are eventually constant
If an integer-valued <monotone sequence> $(a_n)$ is nondecreasing and bounded above by $M$, it can increase strictly at most $\lfloor M\rfloor-a_1$ times. Thus its values eventually remain constant. Such sequences are determined by a finite initial list followed by a constant tail, so their collection is a <countable set>.