Problem with inequality proof with metrics.

by argiriskar   Last Updated October 17, 2018 12:20 PM - source

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 $\frac{1}{d(a,x)}-\frac{1}{d(a,y)} \leq d(x,y) \leq \frac{1}{d(a,x)}+\frac{1}{d(a,y)}$.

Is this provable ?

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

Any help would be appreciated.

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