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I have an Analysis exam coming soon, and found this practice problem a bit challenging. Any help on this would be appreciated.

A metric space $M$ with metric $d$ can always be re-metrized so the metric space becomes bounded. Simply define the bounded metric: $$ \varrho(p,q) = \frac{d(p,q)}{1+d(p,q)} $$ Prove that $\varrho$ is a metric. Why is it obviously bounded?

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  • $\begingroup$ A simpler equivalent bounded metric is $\delta(p,q)=\min\{d(p,q),1\}$. $\endgroup$ – egreg Feb 25 '15 at 0:21
  • $\begingroup$ @David 's answer shows it's bounded and $\rho(p,q)\leq \mathrm{d}(p,q)$. So if you know another bound such that $\alpha\cdot\rho(p,q)\geq \mathrm{d}(p,q)$, then the metrics are equivalent in terms of their open sets $\endgroup$ – snulty Feb 25 '15 at 1:23
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For any $x\geq0$, we have $$\frac{x}{1+x}<1.$$

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