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For a set $X$, let $\mathbb{R}^{X}$ be the set of all maps from $X$ to $\mathbb{R}$.

For $f,g\in\mathbb{\mathbb{R}}^{X}$, define $$d(f,g) = \sup_{x\in X}\frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}.$$

I am trying to show that $(\mathbb{R}^{X},d)$ is a metric space but I can't get the bounds in the right way.

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    What part of showing that $(\mathbb{R}^x , d)$ that is a metric space are you stuck on? What have you tried?2012-11-22
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    What is the connection between $\mathbb R^x$ and $X$? Or do you mean $\mathbb R^X$?2012-11-22
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    Yo can first prove that $\rho(f,g) = \sup_{x\in X} |f(x) - g(x)|$ is a metric. Then prove that if a function $\rho(x,y)$ is a metric then $d(x,y) = \rho(x,y)/ (1 + \rho(x,y))$ is a metric.2012-11-22
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    @Yury These functions may not be bounded, so $\rho$ is not really a metric.2012-11-22

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