Prove that||x|−|y||≤|x−y|Proof of the Triangle Inequality|x+y|≤|x|+|y|.Proof of the reverse triangle inequality:||x|−|y||≤|x−y|.

Alfred Martin

Alfred Martin

Answered

2021-12-16

Prove that ||x||y|||xy|
Proof of the Triangle Inequality |x+y||x|+|y|.
Proof of the reverse triangle inequality:
||x||y|||xy|.

Answer & Explanation

esfloravaou

esfloravaou

Expert

2021-12-17Added 43 answers

|x|+|yx||x+yx|=|y| 
|y|+|xy||y+xy|=|x| 
Move |x| to the right hand side in the first inequality and |y| to the right hand side in the second inequality. We receive
|yx||y||x| 
|xy||x||y|
From absolute value properties, we have that |yx|=|xy|, and if ta  and  ta then t|a|
Putting these two facts together, we get the reverse triangle inequality:
|xy|≥∣∣|x||y|∣∣.

Mason Hall

Mason Hall

Expert

2021-12-18Added 36 answers

Consider |x||y|. Therefore: 
||x||y||=||xy+y||y||||xy|+|y||y||=||xy||=|xy|.

nick1337

nick1337

Expert

2021-12-28Added 573 answers

Since we are thinking about the reals, R, then a field's axioms apply. specially forx,y,zR,x+(x)=0;x+(y+z)=(x+y)+z; and x+y=y+x.
Start with x=x+0=x+(y+y)=(xy)+y.
Then apply |x|=|(xy)+y||xy|+|y|. By so-called "first triangle inequality."
Rewriting |x||y||xy| and ||x|y|||xy|.
The item of Analysis that I find the most conceptually daunting at times is the notion of order (,,<,>), note how some sentences might be improved into shorter versions.

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