Real Analysis triangle inequity

[Askhwhelp]

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

 

[Ray Vickson]

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

Sure: just use your ideas above and see where they lead.

 

[Askhwhelp]

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

 

[Dick]

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

Sure, that's a fine proof.