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.