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I am considering the metric space $(M,d)$, where $M$ is nonempty. I define $\rho:M\times M\rightarrow\mathbb{R}$ by $\rho(x,y):=\dfrac{d(x,y)}{1+d(x,y)}$. I have shown that $\rho$ is also a metric on $M$ and that $d$ and $\rho$ are equivalent. Now I want to show that $M$ is bounded with respect to $\rho$, but I do not really understand how I can show this. Has someone a hint for me?

Falrach
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mathastic
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    Note that $\rho(x,y) < 1$ for all $x,y \in M$. – Charlie Sep 22 '21 at 14:25
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    @Randall your proposed duplicate has no answers showing that the metric is bounded; they all seem to assume it instead. – postmortes Sep 22 '21 at 14:28
  • You are correct: I missed this. Vote to reopen. – Randall Sep 22 '21 at 14:39
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    I've voted to reopen since, as postmortes points out, the linked post doesn't answer this question. I expect this question is a duplicate, but not to any part of that question. – Brian Moehring Sep 22 '21 at 14:41
  • @Charlie, thnx. I thought that this question should be more complex, but I now indeed see that $\rho(x,y)\in[0,1)$ and that M is thus bounded – mathastic Sep 22 '21 at 15:06

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