I need help with advanced calculus

by writing a = (a+b) +(-b) use the triangle inequality to get |a| - |b| <= |a+b|. The interchange a and b to show that ||a| - |b|| <= |a+b|

|a| = |(a + b) + (-b)|
....≤ |a + b| + |-b|, by triangle inequality
....= |a + b| + |b|.

Solving for |a + b| yields |a| - |b| ≤ |a + b|.
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Interchanging a and b yields |b| - |a| ≤ |b + a|
==> -(|a| - |b|) ≤ |a + b|.
==> |a| - |b| ≥ -|a + b|.

Putting these inequalities together yields
-|a + b| ≤ |a| - |b| ≤ |a + b|.

Therefore, we have ||a| - |b|| ≤ |a + b|.

I hope this helps!