I am looking to write a proof (without breaking it into cases) for the following:Prove that |x+y| ≤ |x| + |y| using the following assumptions:-|x| ≤ x ≤ |x|if c ≤ r, then c ≥ -rif r ≥ -c, and r ≤ c, then -c ≤ r ≤ c
Before I go on, I have to apologize. I was unable to come up with a proof of my own (I kept getting stuck), so I searched the internet (this property is famously known as the "Triangle Inequality", and has applications in number theory, calculus, physics, and linear algebra) and found two different proofs that appeared side-by-side on numerous sites. I have copied and pasted these proofs into this post BUT I must warn you that I myself do not believe either of them. Though these proofs seem unanimously accepted by the people of the internet, they do not seem correct to me (I will explain why later). I am doing my best to answer your question, but just know that.
The red question marks indicate areas where the proof fails for me. If you can find an explanation for these areas, let me know.
