[Math] Showing that the set $A = \{(x,y): y < 0\}$ is open

Question

Show that the set $A = \{(x,y): y < 0\}$ is open. I understand that to do this, i need to take an open ball centered at an arbitrary point in $A$ with a positive radius and show it is contained in the set. However, i'm not sure what my radius should be, or how to prove this formally.

Answer 1

Hint: If you've fixed a point $(x, y)$ with $y < 0$, consider the ball with radius $$r = \frac{|y|}{2}$$