I found the following exercise in Chartrand's book at the end of chapter 3, and attempted to prove it, but I am not very satisfied because it does not seem legitimate. I have not officially learned the mathematical proofs so kindly please help me.

Exercise 3.45:

Prove that if and are two positive integers, then

The following is my weak attempt:

Let , and

The algebraic reduction of

We can restate the question as and

Proof:

By trichotomy law, we have three cases, namely and

Case 1: Assume that Since

Case 2: Assume that . Since the square of any integer is a positive integer then and . It follows that

Since Case 3 is similar to case 2, we will not repeat. Q.E.D