I’m a beginner. According to my professor, I must write out every step as clearly as possible.

**Question: Prove the set of even integers is well-ordered using the well-ordering principle.**

**Solution (According to Bartelby.com):** By definition, the set of positive even integers is the subset of positive integers. Moreover, since the even integers are well-ordered, (by the well-ordering principle), the positive even integers are well-ordered.

**However, I’m unable to understand the proof. How do we know that if a set has the property then its subset has the property? (My Professor demands I rely on proofs rather than intuition).**

Below is my attempt at the problem:

**My attempt:** By definition, the set of positive even integers is the subset of positive even integers. Hence if the set of positive even integers are well-ordered, then; the set of postive integers are well-ordered. Despite this, we must prove if positive integers are well-ordered then positive even integers *are well-ordered*. Therefore, we take the contrapositive.

If the positive integers are not well-ordered then the positive even integers are not well-ordered; however, the positive integers are well-ordered (by the well-ordering principle). This is a contradiction to our hypothesis where the positive integers are not well-ordered. Hence if the positive integers are well-ordered then the positive even integers are well-ordered. Therefore, the positive even integers are well-ordered.

**Question: Am I correct or have I made a mistake in my steps? Try and explain step by step?**