Images of Finite Sets Are Finite, Bounded Sets of Natural Numbers Are Finite, and Nonempty Finite Totally Ordered Sets Have Extreme Elements

In the setting of The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion, let finite sets be as in Finite Sets §finite and intervals $\{m,\dots,n\}$ as in Intervals of Natural Numbers §interval, and let $A$ and $B$ be sets.

¶ If $A$ is finite and $f:A\to B$ is a map, then $f(A)$ is finite.

¶ Let $\le$ be a total order on a set $X$. Every nonempty finite subset $S$ of $X$ has a greatest element and a least element, $\max S$ and $\min S$ of Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order §least.

¶ A subset $S$ of $\mathbb{N}_{0}$ is finite if and only if there is $b\in\mathbb{N}_{0}$ with $k\le b$ for every $k\in S$; in particular every interval $\{m,\dots,n\}$ with $m,n\in\mathbb{N}_{0}$ is finite.