Order preserving maps

Question

Suppose $f:X \to Y$ is order preserving. Let $A$ be a subset of $X$. Does is follow that if $A$ is well ordered then $f(A)$ is well ordered?

Answer 1

Yes. If $S$ is a nonempty subset of $f[A]$, let $T=\{x\in A:f(x)\in S\}$. Then $T$ has a least element $t$, and $f(t)$ is clearly minimal in $S$.