What is property of modulas.

Proof of the properties of the modulus Proof of the Triangle Inequality #1: 1. |z1 + z2|<=|z1| + |z2| |z1 + z2|= square 1. |z1| + |z2|= square 2. We have to prove that square 1 square 2 is true. Square both sides. equation square 3. 2x1x2 +2y1y2 <= square 4 Square both sides again. 2x1x2y1y2 <= x12y22 + y12x22 and we get 0<=(y1x2 - x1y2)2. It is true because x1, x2, y1, y2 are all real. Proof of the Triangle Inequality #2: 2. |z1 + z2|>=|z1| - |z2| We have to prove that square 1 square 5 is true. Square both sides. >= - square 4. 2x1x2 +2y1y2 >=-square 4. Multiply both sides by (-1/2). -(x1x2 +y1y2) <=square 1. Square both sides. - x12x22 - 2x1x2y1y2 - y12y22<= x12x22 + x12y22 + y12x22+ y12y22 and we get 0<= (y1x2 + x1y2)2 + 2x12x22 + 2y12y22. It is true because x1, x2, y1, y2 are all real, and squares of real numbers are >=0.