Theorem. $|a| - |b| \leq |a - b|$.
Are the following two proofs equivalent?
Proof I. $|a| - |b|$ $\leq$ $|a| + |b|$ by the triangle inequality. This is equal to $$|a| - |a| - |b|\leq|a| - |a| + |b|$$ which is equal to $$-|b|\leq|b|,$$ which is true. QED.
Proof II. Using the trick that $|a| = |a + b - b|$. Then $$|a| \leq |a -b| + |b|$$ by the triangle inequality and moving $|b|$ to the left side gives us $|a| - |b|\leq$ $|a - b|$.
Now the first one was my way and the second one was the official answer. Is mine considered wrong because my final statement wasn't the statement they wanted me to prove? If so, how would I know that I needed to use this little algebra trick to start my proof? Or can I just write my proof backwards at least?
Just a little confused on some formality, thanks!