How does one prove d ( x , z ) &#x2264;<!-- ≤ --> d ( x , y ) + d (

Damion Hardin

Damion Hardin

Answered question

2022-04-10

How does one prove d ( x , z ) d ( x , y ) + d ( y , z ) for d ( x , y ) = | x y | given x , y , z R ?

Answer & Explanation

Adeline Shah

Adeline Shah

Beginner2022-04-11Added 18 answers

We have
| x y | = | x z + z y | | x z | + | z y |
Ainsley Zimmerman

Ainsley Zimmerman

Beginner2022-04-12Added 3 answers

Firstly, for a , b R we have
| a + b | 2 = ( a + b ) 2 = a 2 + 2 a b + b 2 | a | 2 + 2 | a | | b | + | b | 2 = ( | a | + | b | ) 2
and by taking the square roots we have
| a + b | | a | + | b | .
Then, by using the above result, for x , y , z R we have
| x y | = | ( x z ) + ( z y ) | | x z | + | z y | .
Q.E.D.

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