I am a first year Mathematics student. I have this homework where I have to prove the reverse triangle inequality $(||x|-|y|| ≤ |x-y| )$. I have proven it with the triangle inequality, so I had to include the proof of triangle inequality as well. So I proved it too, but I found no proof like mine, so I am a little paranoid about my proof. I wanted to ask you people about it. The proof:
Let a and b be real numbers.
- Since $a ≤ |a|$ and $b ≤ |b|$, we can get the sum of both inequations. So, $a + b ≤ |a| + |b|$
- And similarly $-a ≤ |-a| = |a|$ and $-b ≤ |-b| = |b|$. So, $(-a) + (-b) ≤ |a| + |b|$ which implies that $-(a + b) ≤ |a| + |b|$.
- Hence, since both $a + b ≤ |a| + |b|$ and $-(a + b) ≤ |a| + |b|$ are true, $|a + b| ≤ |a| + |b|$ is also true as desired. QED
Any kind of help is appreciated.