Intersection of all open intervals of the form $(-\frac{1}{n}, \frac{1}{n})$ is $\{0\}$? In Mathematical Analysis by Apostol he mentions that the "Intersection of all open intervals of the form $(-\frac{1}{n}, \frac{1}{n})$ is $\{0\}$"
Obviously this is a super basic question but I thought that an open interval does not include the endpoints, so from the limit as $n\to\infty$ on both sides we get (0,0), i.e. if $x$ belongs to the intersection then $0<x$ and $x<0$ which no real number satisfies. 

$\cap_{n=1}(-\frac{1}{n}, \frac{1}{n}) = {0}$ is equivalent to the statement that for all $n \in \mathbb{N}, 0 \in (-\frac{1}{n}, \frac{1}{n})$, which is obviously true.
As $n \rightarrow\infty$, $(-\frac{1}{n}, \frac{1}{n})$ approaches the zero, but never reaches the zero
