Suppose $(X,d)$ is a metric space, then question is to prove that  $\rho(x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric.  This question is from Munkres Topology book.  Hint : If $f(x)=\frac{x}{1+x}$ for $x&gt;0$ us mean value theorem to show that  $f(a+b)-f(b)\leq $ some wierd symbol which i assume to be $f(a)$..  Suppose we prove that $f(a+b)-f(b)\leq f(a)$ then $f(a+b)\leq f(a)+f(b)$. We then have $$\frac{a+b}{1+a+b}\leq \frac{a}{1+a}+\frac{b}{1+b}$$   We have $f'(x)&gt; 0$ so, $f$ is increasing...  We have $d(x,y)\leq d(x,z)+d(z,y)$.. So, we have $f(d(x,y))\leq f(d(x,z)+d(z,y))$ i.e., $$\frac{d(x,y)}{1+d(x,y)}\leq \frac{d(x,z)+d(z,y)}{1+d(x,y)+d(z,y)}$$  With $$\frac{a+b}{1+a+b}\leq \frac{a}{1+a}+\frac{b}{1+b}$$   we have  $$\rho(x,y)=\frac{d(x,y)}{1+d(x,y)}\leq \frac{d(x,z)+d(z,y)}{1+d(x,y)+d(z,y)}\leq \frac{d(x,z)}{1+d(x,z)}+\frac{d(z,y)}{1+d(z,y)}=\rho(x,z)+\rho(z,y)$$  So, all that is left to prove is that $f(a+b)-f(b)\leq f(a)$..  Suppose that $a&lt;b$.. By mean value theorem, there exists $c\in (b,a+b)$ such that $$\frac{f(a+b)-f(b)}{(a+b)-b}=f'(c)=\frac{1}{1+c^2}$$ i.e., $$f(a+b)-f(b)=\frac{a}{(1+c)^2}\leq \frac{a}{(1+a)^2}\leq \frac{a}{1+a}=f(a)$$  Last inequality holds because $a&lt;c$  Suppose that $b&lt;a$.. By mean value theorem, there exists $c\in (a,a+b)$ such that $$\frac{f(a+b)-f(a)}{(a+b)-a}=f'(c)=\frac{1}{1+c^2}$$ i.e., $$f(a+b)-f(a)=\frac{b}{(1+c)^2}\leq \frac{b}{(1+b)^2}\leq \frac{b}{1+b}=f(b)$$  Last inequality holds because $b&lt;c$..  So, in any case we have $f(a+b)\leq f(a)+f(b)$ and we are done..  I just want to know if there are any gaps in this solution..
Let $(X,d)$ be a metric space and for $x,y \in X$ define $$d_b(x,y) = \dfrac{d(x,y)}{1 + d(x,y)}$$ a) show that $d_b$ is a metric on $X$  Hint: consider the derivative of $f(t)$ = $\dfrac{t}{1+t}$  b) show that $ d$ and $ d_b $ are equivalent metrics. c) let $(X,d) $ be $(\mathbb{R}, |\cdot|)$ Show that there exists no $ M&gt;0$ such that $ |x-y| $ $\leq$ $Md_b$ $(x,y)$ for all $ x,y$ $\in$ $\mathbb{R}$  I have calculated $f(t)$ and $f'(t)$ and from this I know $f(t)$ is an increasing function as $f'(t)$ is strictly positive. But I don't know where to go from here or how to do parts b) and c)
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