# About compactness and completeness of set $X = \mathbb{R}$ with the metric $d(x, y) = \frac{|x-y|}{1+|x-y|}$

What can we claim concerning the density and also efficiency of the set $X = \mathbb{R}$ with the statistics $$d(x, y) = \frac{|x-y|}{1+|x-y|}\;?$$

I attempted by revealing that $d(x, y)<1$ and also hence it is bounded. Yet just how to show remainder of the points? Can any person aid me? Many thanks

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2022-07-25 20:46:18
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