Ellipse is Bounded in Plane

Theorem
Let $E$ be an ellipse embedded in in a Euclidean plane.

Then $E$ is bounded.

Proof
Let a Cartesian coordinate system be applied to the Euclidean plane in which $E$ is embedded.

Let $E$ be expressed in reduced form:

Hence:
 * there are no real values of $x$ for $\size y > b$
 * there are no real values of $y$ for $\size x > b$.

Hence $E$ exists entirely within the rectangle whose sides are $x = \pm a$ and $y = \pm b$.

The result follows.