For the set $$S=\left \{ (x,y) \in \mathbb{R}^2 : x^2+2y^2=22 \right \}$$ this is an ellipse and it looks like this:
[![Ellipse](https://i.stack.imgur.com/ysk0k.png)](https://i.stack.imgur.com/ysk0k.png)

I need to show that it is compact which means it is bounded and closed.
What is formal way to show compactness?