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Let $a,b \in \mathbb{R}$. Let $a=(a+b)+(-b)$, show that $|a|-|b| \leq |a+b|$, using the Triangle Inequality.

This is currently what I have done. I think I am going about this in a wrong way though.

Let $a,b \in \mathbb{R}$, such that $a=(a+b)+(-b)$. Then by the Triangle Inequality, we have, $|(a+b)+(-b)+b| \leq |(a+b)+(-b)| + |b|$. Then $|(a+b)+(-b)+b| \leq |a+b| + |-b + b| \leq |(a+b)+(-b)| + |b| \leq |a+b| + |-b| + |b|$.

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What have you tried? I mean, it is quite obvious to what you should apply the triangle inequality, and you're done right after that. – TMM Sep 19 '11 at 0:16
up vote 3 down vote accepted

HINT: $|a| = |(a + b) + (-b)| \leq |a + b| + |-b| = |a + b| + |b|$. Now subtract $|b|$ from both sides...

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