Non-empty finite point set is closed Subset of $\Bbb R^2$:
My book says that non-empty finite point sets are closed. Why is this?
Since it is a finite point set, it necessarily has no limit points within it, since every neighborhood of a limit point has infinite many points in it.
So a nonempty finite point set,$E$, cannot have a limit point, and this implies that all limit points of $E$ are in $E$?

Suppose $x_n \to x$, with $x_n \in E$, where $E$ is finite. Let $L = \{e  \in E | x_n = e \text{ infinitely many times}\}$.
Since the limit is unique, we see that $L$ is a singleton, that is $L=\{e\}$ for some $e$. Then we see that we must have $x_n = e$ for all $n \ge N$, where $N$ is some index. Hence $x \in E$ and so $E$ is closed.
