The intersection of infinitely many open sets is closed (Proof for specific example)

Question

An example where the intersection of infinitely many open sets is closed is the sets $$(-1,1),(-\frac{1}{2},\frac{1}{2})...(-\frac{1}{n},\frac{1}{n}).$$ But how can I prove that it is indeed closed?

Answer 1

Try to explicitely determine the intersection of these sets. It turns out to be a singleton, thus closed in the Euclidean topology on $\mathbb{R}$.

Answer 2

For all $n$, $0 \in (-\frac{1}{n}, \frac{1}{n})$ for all natural number $n \ge 1$. So $0 \in \bigcap_{n \ge 1} (-\frac{1}{n}, \frac{1}{n})$

If $x \neq 0$, we know that $|x|>0$ and so for large $N$ we have that $\frac{1}{N} < |x|$, (essentially as $\frac1n \to 0$ as $n \to \infty$) and thisimplies that $x \notin (-\frac{1}{N}, \frac{1}{N})$ and so $x \notin \bigcap_{n \ge 1} (-\frac{1}{n}, \frac{1}{n})$.

Together we see that $\bigcap_{n \ge 1} (-\frac{1}{n}, \frac{1}{n}) = \{0\}$ which is closed, e.g. because $\mathbb{R}\setminus\{0\} = (-\infty,0) \cup(0,\infty)$ which is open as the union of two open intervals.