What are the Cauchy sequences in $(X,d)$ when $X = \mathbb{R}$ and d represents the discrete metric?

Question

I know that a discrete metric $d$ on $X$ is defined by

$$d(x,y) = \begin{cases} 1 &\text{if $x \neq y$} \\ 0 & \text{if }x = y. \end{cases}$$

for any $x,y \in X$.

But what exactly are the Cauchy sequences in a metric space $(X,d)$ when $X = \mathbb{R}$ and $d$ represents the discrete metric?

Answer 1

Suppose that $(x_n)$ is a Cauchy sequence in $(X,d)$. Then there is $N \in \mathbb N$ such that

$$d(x_n,x_m)< 1/2$$

for $n,m \ge N.$ But this means $d(x_n,x_m)=0$ for $n,m \ge N.$ Hence

$$x_n=x_N$$

for $n,m \ge N.$