Generalized Well-Ordering Principle Question

I'm trying to do this proof, but I'm coming across questions about certain properties of well ordered sets that aren't answered by my material. Here's the problem I'm working on:

For any fixed integer $\displaystyle m$, prove that the set $\displaystyle X = \{a \in \mathbb{Z} : m \leq a\}$ is well-ordered.

In trying to prove this, but I've come across a question:

For any non-empty subset of $\displaystyle X$, $\displaystyle Y$, we shall consider the set $\displaystyle S = \{b - m : b \in Y\}$. Regardless of the set $\displaystyle Y$, $\displaystyle S \subseteq \mathbb{N} \cup \{0\}$, and thus $\displaystyle S$ is well-ordered (we have proven that the subset of a well-ordered set is well-ordered). Does this imply that $\displaystyle Y$ 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.