A property on some unbounded metric spaces

Question

Suppose that $(X,d)$ is a metric spaces. Which condition(s) can guaranties the following property: $\forall x, \forall y \in X, \exists \{z_n\}$ such that $\lim_{n\to +\infty } d(x,z_n)=+\infty$ and $\lim_{n \to +\infty } \frac{d(x,z_n)}{d(y,z_n)}=1 $.

Note: I should say, I know this property is true for some spaces like quasi normed spaces, and I want to know more about other spaces with this property.

Answer 1

Unboundedness guarantees that there is one sequence $(z_n)_n$ such that $d(x,z_n)\to\infty$ for all $x$. That sequence also satisfies the second requirement via the triangle inequality: $$ \frac{d(x,z_n)}{d(y,z_n)}\le \frac{d(x,y)}{d(y,z_n)}+1 $$ and symmetrically $$ \frac{d(y,z_n)}{d(x,z_n)}\le \frac{d(y,x)}{d(x,z_n)}+1 $$ which shows that the limit of the quotient is equal to $1$