how to prove that any Cauchy sequence in a discrete space is stationary
Let $(x_n)$ be a cauchy sequence then $$\forall \varepsilon>0, \exists n_0\in \mathbb{N},\forall p,q \geq n_0\Rightarrow \begin{cases} 1\leq \varepsilon, x_p\neq x_q\\ 0\leq \varepsilon,x_p=x_q\end{cases}$$
how to continue?
thank you
The function $d:\mathbb N\times \mathbb N\rightarrow[0,+\infty)$ defined as $d(m,n)=\left|\displaystyle\frac{1}{m}-\frac{1}{n}\right|$ is a metric, which induces discrete topology on $\mathbb N$. The sequence $(n,n\in\mathbb N)$ is a Cauchy sequence in the space $(\mathbb N,d)$, and this sequence is not stationary.