prove that ||x+y|| less than or equal to ||x|| + ||y||

Question

Prove and state when the equality sign is attained

Answer 1

|x+y|: let x,y>=0, then |x+y|=|x|+|y|.

Let x,y<0, |x+y|=-(x+y); |x|=-x, |y|=-y; so |x+y|=-x-y=|x|+|y|.

Let x<0, y>0 and x+y>0, so |x+y|=x+y; |x|=-x and |y|=y, x+y<y-x (x is negative so -x is positive). |x+y|<|x|+|y|.

Let x<0, y>0 and x+y<0, so |x+y|=-x-y; |x|=-x and |y|=y, -x-y<y-x. |x+y|<|x|+|y|.

The symmetry of the expressions means that x and y are interchangeable, so |x+y|<=|x|+|y|.

Comments

Does the sign

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Thank you very much for your help, Rod.