So I came upon this issue when trying to prove the following inequality:
|x| - |y| $\le$ |x - y|
one of the triangle inequalities.
So one way I thought of proving this is to square both sides. With a few simple steps, the inequality simplifies to |x||y| $\ge$ xy , which we know is true. But is this enough to really prove this inequality?
The other way to prove it would be with contradiction; ie, "Just suppose that |x| - |y| > |x - y| " and then show that this simplifies to a false inequality.
Are these two different approaches comparable in how deeply they prove the inequality?
You basically ask if between two different kind of proofs one is "better" or "more valid" than the other. The answer is No. Proof is proof: As long as the reasoning is logically correct, a proof proves the statement it is a proof of. A proof by contradiction is as good as any other.
On the other hand, different proofs for the same statement (regardless if one of them is by contradiction or not) can provide different insight into a problem or statement, but that is subjective and not measurable.
EDIT: I should mention that there is a philosophical branch of mathematics called "Constructivism", where proof by contradiction and axiom of choice are not allowed. If there are merits in that philosophy, I let for you to decide. However, the majority of the mathematical doesn't feel obliged to follow the constructionistic ways and will accept proofs by contradiction without problem.