- #1
Somefantastik
- 230
- 0
Hey folks, can someone quickly check my algebra?
Given:
[tex] d(x,A) \leq d(x,y) + d(y,A) [/tex]
To show:
[tex] \left|d(x,A) - d(y,A) \right| \leq d(x,y) [/tex]
Proof:
from given, [tex] d(x,A) - d(yA) \leq d(x,y); [/tex]
and
[tex] -d(x,A) + d(y,A) \geq -d(x,y); [/tex]
[tex]\Rightarrow d(y,A) - d(x,A) \geq -d(x,y); [/tex]
[tex]\Rightarrow -d(x,y) \leq d(x,A)-d(y,A) \leq d(x,y); [/tex]
Therefore
[tex] \left|d(x,A) - d(y,A) \right| \leq d(x,y) [/tex]
Given:
[tex] d(x,A) \leq d(x,y) + d(y,A) [/tex]
To show:
[tex] \left|d(x,A) - d(y,A) \right| \leq d(x,y) [/tex]
Proof:
from given, [tex] d(x,A) - d(yA) \leq d(x,y); [/tex]
and
[tex] -d(x,A) + d(y,A) \geq -d(x,y); [/tex]
[tex]\Rightarrow d(y,A) - d(x,A) \geq -d(x,y); [/tex]
[tex]\Rightarrow -d(x,y) \leq d(x,A)-d(y,A) \leq d(x,y); [/tex]
Therefore
[tex] \left|d(x,A) - d(y,A) \right| \leq d(x,y) [/tex]