Real Analysis triangle inequity

  • Thread starter Askhwhelp
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  • #1
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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
 

Answers and Replies

  • #2
Ray Vickson
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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.
 
  • #3
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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
 
  • #4
Dick
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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.
 

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