How to prove a set is open by finding a ball around each point

Question

How can we show that the set $ U=\{(x,y) \text{is an element of } \mathbb{R^2}:x^2+4y^2<4\}$ is open by finding a ball around each point which is contained in U

This question has been asked before but not answered.

Answer 1

For each point $(x_0,y_0)$ of $U$ we have $$x_0^2+4y_0^2=4-\epsilon$$for some $\epsilon>0$. Therefore $B_\epsilon(x_0,y_0)=\{(x,y)\in\Bbb R^2|(x-x_0)^2+(y-y_0)^2=\dfrac{\epsilon^2}{100}\}$ has been contained in $U$