$\mathbb{R}$ is endowed with the metric $d(x,y)=|\arctan(x)-\arctan(y)|$ ,
i want to prove that $(x_n=n)$ is a Cauchy sequence, i see that $\lim_{p,q\rightarrow\infty} d(x_p,x_q)=0$ then $(x_n)$ is a cauchy sequence, but if i want to find $n_0$ , how to do for $|\arctan(p)-\arctan(q)|<\varepsilon$
i mean how to prove using the definition:$$\forall\varepsilon>0, \exists n_0\in \mathbb{N}, \forall p,q\in \mathbb{N}, p>q\geq n_0\Rightarrow d(x_p,x_q)<\varepsilon$$ we have
$|\arctan(p)-\arctan(q)|<|\arctan(p)|+|\arctan(q)|\leq 2 |\arctan(p)|<\varepsilon$
how to find $n_0$ ?
Thank you