The discussion revolves around proving the triangle inequality for real numbers, specifically the inequality |x| + |y| ≤ |x+y| + |x-y|. Participants are exploring various approaches to establish this inequality.
Some participants have shared their reasoning and steps taken, while others have provided affirmations of the approaches discussed. The conversation reflects an ongoing exploration of the proof without reaching a definitive conclusion.
There is a repetition in the posts, indicating a potential misunderstanding or a need for clarification on the proof process. The participants are working within the constraints of a homework assignment, seeking guidance without explicit solutions.
Askhwhelp said:Prove |x|+|y| ≤ |x+y| + |x-y| for all real numbers x and y.
Some ideas I have is let a = x+y and b = x-y and apply triangle inequity
Could anyone give me some direction?
Thanks
Askhwhelp said:let a = x+y and b = x-y.
|2x| = |a+b| <= |a| + |b|
|2y| = |a- b| <= |a| + |b|
So |2x| + |2y| <= 2(|a| + |b|)
Divide both sides by 2, we get
|x|+ |y| <= |a| + |b|
Is this right way?
Thanks