# Generalized Well-Ordering Principle Question

For any fixed integer $m$, prove that the set $X = \{a \in \mathbb{Z} : m \leq a\}$ is well-ordered.
For any non-empty subset of $X$, $Y$, we shall consider the set $S = \{b - m : b \in Y\}$. Regardless of the set $Y$, $S \subseteq \mathbb{N} \cup \{0\}$, and thus $S$ is well-ordered (we have proven that the subset of a well-ordered set is well-ordered). Does this imply that $Y$ is well-ordered? I can't seem to find any information in my class materials that show this to be true, and I can't really see how to prove that property.