Consider
$$d(x,y)=\frac{2\|x-y\|}{(1+\|x\|^2)^{1/2}(1+\|y\|^2)^{1/2}},\hspace{5mm}x,y\in \mathbb{R}^n.$$
$d$ is a metric in $\mathbb{R}^n$ known as chordal metric. I want to show that this metric is topologically equivalent to the Euclidean metric $\|x-y\|$.
In order to do this, I tried to find positive real constants $M,N$ such that
$$M\cdot d(x,y)\leq \|x-y\|\leq N\cdot d(x,y),\forall x,y\in \mathbb{R}^n.$$
It's trivial to see that
$$\frac{1}{2}d(x,y)\leq \|x-y\|$$
finding, for instance, $M=1/2$. But I don't know how to find such $N$ as above!
Any idea? Thank you!
Hint: You're not going to find the number $N$ independent of $x,y$. That's easy to see: take $x,y$ at a very, very large distance from the origin (making the denominator large), and satisfying $||x-y||=1$.
However, you don't need something that strong, because $N$ can be allowed to depend on $x$ and $y$ in some fashion. For instance, it would certainly suffice if for all $x$ you could find $N_x$ such that for all $y$ in a ball of radius $1$ around $x$, the second inequality holds using the constant $N_x$.