Let $\mathcal U$ be an open and bounded subset of $\mathbb R^2$.So I have two questions:
Is there $r \gt 0$ such that $xy \le r^2\;\;\;\;\forall (x,y) \in \mathcal U$ ? (I mean there's for sure $r \gt 0$ such that $(x-x_0)^2 + (y-y_o)^2 \le r^2\;\;\;\forall (x,y),(x_0,y_0) \in \mathcal U\;\;$ since $\mathcal U$ is open but could somebody find $r \gt 0$ with this property?) If there isn't such $r$, what would be the upper bound then?
Could I choose $\varepsilon \gt 0$ as large as I want in order the product $\varepsilon xy$ would pe positive no matter what the sign of $xy\;is\;\;\;\forall (x,y) \in U$?
I'm new to topology/analysis so I would appreciate any help.. Thanks in advance
The answer to the first question is trivially yes. Just choose $R:=r+\sqrt{x_0^2+y_0^2}$, then $\mathcal{U}\subset B_R(0)$, and for any $(x,y)\in B_{R}$ it holds $x,y\leq R$
To answer the second question: if $\epsilon>0$ then the sign of $xy\epsilon$ is nothing but the sign of $xy$.