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 , prove that the set is well-ordered.

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

For any non-empty subset of , , we shall consider the set . Regardless of the set , , and thus is well-ordered (we have proven that the subset of a well-ordered set is well-ordered). Does this imply that 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.