Understanding the class of all convergent sequences in a discrete space. [duplicate]

Prove: $X$ with the discrete metric $d$ where,

$d(x,y)=\begin{cases} 1,&x\ne y\\0,& x=y\end{cases}$

$ (x_n)$ is convergent if and only if it is constant for a sufficiently large $ n$.

I figured that the best way to prove this is by finding a/the Cauchy sequence(s) since that would prove it is convergent, I am just unsure of how to fully complete this as we have not touched on this in depth in the class as of yet.

Please note: This question is for practice. Not part of an assignment for anything for marks. $d(x,y) $ is supposed to be formatted like a piecewise function, I just do not know how to do that formatting in math stack exchange. Thank you for any help.

$(X, d_{discrete}) $ be a discrete space.

A sequence $(x_n) $ in $(X, d) $ is convergent iff $(x_n) $ is eventually constant.

$(x_n) $ is eventually constant if $\exists N\in\mathbb{N}$ such that $ x_n= constant $ [for all $n>N$]

Proof: If $(x_n) $ is eventually constant then it must converge and converges to the constant term.

Now if $(x_n) $ is convergent to $x$ in $(X, d) $

Then given any $\epsilon >0$ $\exists N\in\mathbb{N}$ such that $d(x_n , x) <\epsilon $ for all $n>N$

Since the above inequality is true for any $\epsilon $ as long as it is positive, we can set $\epsilon =1$

Then, $d(x_n, x) <1 , \forall n>N$

Since, $d$ is discrete metric, only possible choice is $d(x_n , x) =0$ for all $n>N$

Hence, $x_n = x $ for all $n>N$

And the sequence is of the form :

$\{x_1 ,x_2 , ...,x_N,x, x, x,...\}$

And hence $(x_n ) $ is eventually constant.