How do you prove this consequence of the triangle inequality?

  • #1

Main Question or Discussion Point

I should know how to do this, but I just can't figure it out. Should be a piece of cake. How do you prove, for [itex]x,y\in \mathbb{R}^n[/itex]:

[tex]
\left| (||x|| - ||y||) \right| \leq ||x-y||
[/tex]
 

Answers and Replies

  • #2
arildno
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We have:
[tex]||x+y||\leq{||}x||+||y||[/tex]

Now, set x=z-y

Then, we get:
[tex]||z||\leq{||z-y||}+||y||[/tex],
For completely arbitrary z and y.

utilize this to derive the desired relation, i.e:
[tex]-||u-v||\leq{||u||}-||v||\leq{||}||u-v||[/tex]
 
  • #3
102
1
I should know how to do this, but I just can't figure it out. Should be a piece of cake. How do you prove, for [itex]x,y\in \mathbb{R}^n[/itex]:

[tex]
\left| (||x|| - ||y||) \right| \leq ||x-y||
[/tex]
||x||=||x-y+y||[tex]\leq[/tex]||x-y||+||y|| ======>

||x||-||y||[tex]\leq[/tex]||x-y||............................................................1

||y||= ||y-x+x||[tex]\leq[/tex]||x-y||+||x||======>

||y||-||x||[tex]\leq||x-y||[/tex] =======>

||x||-||y||[tex]\geq[/tex]-||x-y||............................................................2

from (1) and (2) we get:


-||x-y||[tex]\leq ||x||-||y||\leq ||x-y|| \Longleftrightarrow|(||x||-||y||)|\leq||x-y||[/tex]
 

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