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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!

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    $\begingroup$ ... your first one is wrong because you didn't even prove what you were trying to prove. $\endgroup$ – Eevee Trainer Dec 4 '18 at 4:59
  • $\begingroup$ yeah I kind of realized when I was done that it wasn't correct. But how would someone figure out the solution to the second one? Like how do I know that when I see that question I need to go "OK let's use the fact that a = a + b - b? $\endgroup$ – ming Dec 4 '18 at 5:13
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    $\begingroup$ @ming: it’s just mathematical maturity. You shouldn’t just “know” it, but it is something you should come to recognize given enough exposure and practice. You want to be able to recognize your first attempt as being wrong sooner rather than later to aid in understanding. $\endgroup$ – Clayton Dec 4 '18 at 6:17

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