Subset of Euclidean Plane whose Product of Coordinates are Greater Than or Equal to 1 is Closed

Theorem
Let $\struct {\R^2, \tau_d}$ be the real Euclidean plane.

Let $A \subseteq R^2$ be the set of all points defined as:
 * $A := \set {\tuple {x, y} \in \R^2: x y \ge 1}$

Then $A$ is a closed set in $\struct {\R^2, d}$.

Proof
By definition, $\tau_d$ is the topology induced by the Euclidean metric $d$.

Consider the complement of $A$ in $\R^2$:
 * $A' := \R^2 \setminus A$

Thus:
 * $A := \set {\tuple {x, y} \in \R^2: x y < 1}$

Let $a = \tuple {x_a, y_a} \in A^2$.

Let $\epsilon = \size {1 - x_a y_a}$.

Then the open $\epsilon$-ball of $a$ in $\R^2$ lies entirely in $A'$.

As $a$ is arbitrary, it follows that any such $a$ has an open $\epsilon$-ball of $a$ in $\R^2$ which lies entirely in $A'$.

Thus, by definition, $A'$ is open in $\R^2$.

So, also by definition, $A$ is closed in $\R^2$.