If I have two distance functions $d',d$ on a set $X$ and I want to show that the Bi-Lipschitz condition is not satisfied by these distance functions, then how do I do so? I am confusing myself when it comes to negating the quantifiers in the statement.
Statement: $d$ and $d'$ satisfy the condition when there exists constants $\alpha, \beta > 0$ s.t $\alpha \leq d(x,y)/d'(x,y) \leq \beta$ for every $x,y \in X$ with $x \neq y$
For all $\alpha,\beta>0$ there exist $x_,y\in X$ such that $$ \frac{d(x,y)}{d'(x,y)}<\alpha\quad\text{or}\quad\frac{d(x,y)}{d'(x,y)}>\beta. $$