$|.|$ and $d'$ are not equivalent (in metric sense)? Please i need a counter example why $|.|$ and $d'$ are not metricly equivalent where: 
$$d'(x,y)=\left\{
\begin{array}{cc}
  0, & x=y, \\
  |x|+|y|, & x\neq y.
\end{array}\right.$$
Thank you very much .

If $d$ and $d'$ are equivalent, then for each $x,y \in \mathbb{R}$ there are constants $c$ and $c'$ such that $cd(x,y)\leq d'(x,y) \leq c'd(x,y)$. For any $x\not=y \in \mathbb{R} \ :\ d(x,y) = |x-y| \leq |x|+|y| = d(x,y)$, we need only show that no $c'$ can exist. Assume the contrary. Choose an integer $N' > c'$. Then, $d'(N',N'+1)=2N'+1 > c' = c'd(N',N'+1).$ Therefore, $c'$ can't exist.
