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> Let $A= \Bbb{Z} \times \Bbb{R} \subset \Bbb{R}^2$ and let $d\_A$ be the Euclidean metric restricted to $A$. Let $U=\Bbb{Z} \times (-\frac12,\frac12) = \{(m,y) \mid m\in \Bbb{Z}, |y| < \frac12 \}$. Show that $U$ is an open set in the metric space $(A, d\_A).$
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My idea was to define $U\_0 = \{(0,y) \mid |y| < \frac12\}$ and show that this is open and since an arbitary union of open sets are open I would have that $U$ is open since $U = \bigcup\_{i\in \Bbb{Z}}U\_i$. I cannot seem to be able to show that $U\_0$ would be open.

If I pick a point $x=(0, x\_0) \in U\_0$ and construct a ball $B(x,r)$ with $r = \frac12-x\_0 >0$ I should be using this to show that $B(x,r) \subset U\_0$? I'm getting very confused about what should be a subset of what etc in order to get this to work...