Problem with inequality proof with metrics.

Question

In the unbounded metric space $(S,d)$, we fix $a\in S$, and define $$f(x)=\begin{cases} \frac{1}{d(a,x)}, &x\ne a \\ 1, &x=a. \end{cases}$$

I need to prove that $f(x)-f(y)\leq d(x,y) \leq f(x)+f(y) $.

Is this provable ?

The right-hand-side of the inequality seems more true than the left.

Any help would be appreciated.

Answer 1

Taking the standard metric over $\mathbb R$, for any $a$, we can fix $x=a+0.01$ and $y=a-1$ to get

$$\frac{1}{d(x,a)} - \frac{1}{d(a,y)} = 100 - 1 = 99$$

while $d(x,y) = 1.01$.