Intersection of sets within space [0,1/i] vs (0,1/i) as i approaches infinity

Question

Here's an example from All of Statistics by Wasserman

So when A includes 0, the intersection is just 0, but when it only approaches 0, the intersection is the empty set.

Why is this? Is there a name to describe this property, so I can read more about it? I can visualize why, but I'd like to know the formal terms to describe it.

Answer 1

In the second case, for any positive number $x$ pick an $i>\frac{1}{x}$. Then it will follow that $\frac{1}{i}<x$ so $x$ cannot lie in the intersection since in does not lie in $\left(0,\frac{1}{i}\right)$. So no positive number lies in the intersection.

In the first case, $0$ lies in each set, so it lies in the intersection. But by the argument above, no positive $x$ can lie in the intersection.