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 $|(a+b)+(-b)+b| \leq |a+b| + |-b + b| \leq |(a+b)+(-b)| + |b| \leq |a+b| + |-b| + |b|$.