I'm having trouble proving that a given subset is open. Let's say I'm asked to prove the following set is open. $$A =\{(x,y) : -1 < x < 1, -1 < y < 1\}$$ let $(x_0,y_0) \in A$, then $ |x_0| < 1\;and\; |y_0| < 1$. Then I define $$ r = min\{1-|x_0|,1-|y_0|\}$$
Now I need to show that $D_r(x_0,y_0)\subset A$
So I let $(x,y) \in D_r(x_0,y_0)$ and then the distance between the two points is: $$\sqrt{(x-x_0)^2 + (y-y_0)^2}$$ Now I'm stuck. What exactly do I have to show? Can I say that this distance is less than $r$? How do I deal with the fact that $r$ can be one of two values? What is the general approach to such problems?