Show that $(\Bbb N,d_1)$ and $(\Bbb N,d_2)$ are equivalent where $d_1$ is the discrete metric and $d_3(m,n)=|\frac{1}{m}-\frac{1}{n}|$.
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In order to show that $(\Bbb N,d_1)$ and $(\Bbb N,d_2)$ are equivalent,take any $x\in \Bbb N$ and consider $B_{d_1}(x,r)$ .We need to find $s>0$ such that $x\in B_{d_3}(x,s)\subset B_{d_1}(x,r)$ and vice versa.
If we choose $s=r$ in the first case then if $y\in B_{d_1}(x,s)\implies d_1(x,y)<s\implies |x-y|<s\implies |\frac{1}{x}-\frac{1}{y}|<s\implies y\in B_{d_3}(x,r)$.
But I am stuck in the other subset inequality.Please help .
Is there any other way to solve this ?