prove the equation

Question

prove that | |a| - |b| | <= |a-b| for every a,b belongs to real numbers

Answer 1

If a and b are both positive then |a|-|b|=a-b, making ||a|-|b||=|a-b|.

Let a=-m² and b=n², then a-b=-m²-n²=-(m²+n²) and |a-b|=m²+n².

|a|=m², |b|=n², so |a|-|b|=m²-n² and |m²-n²|<m²+n², (difference is less than the sum), therefore:

||a|-|b||<|a-b|.

Let a=m² and b=-n², then a-b=m²+n²;

|a|-|b|=m²-n², |m²-n²​|<m²+n², and ||a|-|b||<|a-b|.

Let a=-m² and b=-n², a-b=-m²+n², |a|=m², |b|=n²;

||a|-|b||=|m²-n²|, |a-b|=|-m²+n²|.

These two quantities have the same value so ||a|-|b||=|a-b|.

Therefore for all a, b ||a|-|b||≤|a-b|.