I have some trouble in proving this statement.
In any topological space, a sequence converges if and only if it is eventually constant.
I struggle to show the direction to the right.
$(\Rightarrow)$ In a topological space $(X,\tau)$, a sequence $(x_n)\in X$ converges to $x\in X$ if for any open set $U$ there exists an $N$ such that $n>N$ implies $x_n\in U$. We want to show there exists $x\in X$ and $N$ such that $n>N$ implies $x_n=x$.
Suppose $(x_n)$ converges to $x$ and let the $x$ we about to show be the limit in the convergent sequence. Then how are we going to find the $N$? If we use the $N$ in the condition convergent sequence then we cannot guarantee $x_n=x$ for all $n>N$. Also should we pick a specific open set in the first place?
Could someone please give some help? Thanks.