Hey folks, can someone quickly check my algebra?(adsbygoogle = window.adsbygoogle || []).push({});

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]

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# Absolute value

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