Prove that the set $\{(x,y)\in \mathbb{R}^2 : x+y\geq0\}$ is closed using sequences. I can intuitively realize why this is true. All the points that are on the line $y=-x$ are in the set. How can I prove this with sequences?

Let $a = (x,y)$ is an accumulation point of the set. Then there exists a sequence $(a_n) \in \mathbb{R}^{2}$ converging to it. Assume that $a = (x,y)$ isn't in the set. Then we have that $x+y = N < 0$. Now take a ball with radius $\frac{|N|}{2}$ around $a$. Obviously that the intersection with the set is empty, but this is impossible, as $a$ is a limit point of a sequence in the set. Hence, by contradiction $a$ is in the set. Therefore as the set contains all of it's accumulation points it's closed.
