Question

For x and y in R, let d(x,y)=(|x-y|)/(1+|x-y|),prove that d defines a.bounded metric on R

Answer 1

1)

Positivity: $d(x,y)\ge0$ with equality if x=y

2)

Symmetry: $d(x,y)=d(y,x)$

$\frac{|x-y|}{1+|x-y|}=\frac{|y-x|}{1+|y-x|}$

3)

Triangle Inequality: $d(x,y)\le d(x,z)+d(z,y)$

$d(x,y)=\frac{|x-y|}{1+|x-y|}=\frac{|(x-z)+(z-y)|}{1+|(x-z)+(z-y)|}\le \frac{|(x-z)+(z-y)|}{1+|(x-z)+(z-y)|}=d(x,z)+d(z,y)$