How can I tell whether this set is closed or open? the question I'm having trouble with is this:
$$A:=\bigcup_{n=1}^{\infty}\left({\left[\frac{1}{n+1},\frac{1}{n}\right) \times \left(0,n\right)}\right).$$
In the Euclidean space $\mathbb{R}^2$, is the subset $A$ either closed or open?
I drew a picture in $\mathbb{R}^2$ and I really think it's open, but I can't come up with an elaborate way to describe why. Any help would be appreciated.
Thanks.

It's open. To see this, note that the only problematic points are $(\frac1{n+1}, y)$ for $y \in ]0,n[$. But $]\frac1{n+2}, \frac1{n}[ \times ]y-\epsilon, y+\epsilon[ \subset$ of the original set, where $\epsilon = \min\left\{ \frac{n-y}{2}, \frac{y}{2} \right\}$.
