Metric such that the diameter of a set is always less than or equal to 1

Question

Given a metric d, we wish to modify it so that the diameter of any set will be bounded by above from 1. How can one find such a metric which works for every set?

Answer 1

Let $d'(x,y) = \min\{d(x,y),1/2\}$.

Answer 2

Yet another method: $d'(x,y)=\frac{d(x,y)}{1+d(x,y)}$. This has the advantage that one can reconstruct the original metric by the formula $d(x,y)=\frac{d'(x,y)}{1-d'(x,y)}$. (Note that $d'(x,y)<1$ for any $x$ and $y$.)