Limits in cofinite topology/why is the limit of x_n = n equal to 1 in the cofinite topology.

Question

Just reading about topological spaces for my exam, and I was wondering if anybody could explain exactly how limits work in the cofinite topology. So I am aware of the topological definition of a limit:

$ Let~(X, \tau)$ be a topological space, and let $x_n$ be a sequence in $X$. x_n is convergent if $\exists L$ s.t.$\forall~U~\in~\tau$, with $L \in U, \exists n\geq N$ s.t. $n\geq N \implies x_n \in U. $

I just can not see how to apply this to the cofinite topology. My lectures claim that $x_n = n \rightarrow 1. $ I have looked elsewhere for answers, but I can't really grasp what they are trying to say. Hopefully, explaining this example will make it clear to me.

Thanks, MSE!

Answer 1

Take any open neighbourhood $U$ of $1$. Since $\Bbb R\setminus U$ must be finite, $U$ contains all but a finite number of terms of the sequence $x_n$. Therefore $x_n\to 1$. You can see $1$ is not special at all.