U in R^2 is open under the Euclidean metric iff U is open under the product metric

So trying to go forward

I have an open ball in $\displaystyle \mathbb{R}^2$ under the Euclidean metric. $\displaystyle u \in U$ implies $\displaystyle \exists$ r s.t. B(u,r)$\displaystyle \subset$ $\displaystyle U $

Open balls with this metric look like circles.

Pick a point in B(u,r) (not necessarily u) say v = $\displaystyle (x_1,y_1)$

I want to produce an open rectangle/square around this point contained in the open ball.

I can draw a picture, so I know this works. But I am not sure how big I can make the rectangle/square.

the distance to the edge of the circle is either $\displaystyle \leq$ r-$\displaystyle y_1$ if going up/down and $\displaystyle \leq$ r-$\displaystyle x_1$ if going left/right.

I dont know how to quantify the radius of the open ball in the product metric (i.e the square/recentagle)

Edit: My guess for the radius was $\displaystyle \frac{1}{2}$ min(r-$\displaystyle y_1$,r-$\displaystyle x_1$)

but i still dont know if this is small enough.

Could any one help me? is there some really easy distance I'm not considering?

Also is going from square to circle any easier?

Thank you for your help.