Negation of $\forall x \in U,$ $\exists r\gt0$ such that $\forall y \in \Bbb R, |y-x|\lt r\Rightarrow y \in U$ $\forall x \in U,$ $\exists r\gt0$ such that $\forall y \in \Bbb R, |y-x|\lt r\Rightarrow y \in U$
Negation: 
$\exists x \in U,$ $\forall r\gt0$ such that $\forall y \in \Bbb R, |y-x|\lt r\land y \notin U$
So it is that right?
 A: The correct one should be
$\exists x \in U,$ such that $\forall r\gt 0$, there exists $ y \in \Bbb R, |y-x|\lt r$ but $y \not\in U$.
A: I'm going to rewrite your formula as:
$$
\forall x \exists r \forall y : x \in U \to \bigg[ r > 0 \land \big[r \in \Bbb R \to (|y-x|\lt r \to y \in U) \big] \bigg]
$$
to better isolate the quantifiers (this follows from the rules around bounded quantifiers).  Now, the negation is just:
$$
\neg \Bigg[ \forall x \exists r \forall y : x \in U \to \bigg[ r > 0 \land \big[r \in \Bbb R \to (|y-x|\lt r \to y \in U) \big] \bigg] \Bigg]
$$
Using $\forall x \phi \equiv \neg \exists x \neg \phi$ we can "move" the negation past the quantifiers:
$$
\begin{align}
\neg \forall x \exists r \forall y & \equiv \\
\exists x \neg \exists r \forall y & \equiv \\
\exists x \forall r \neg \forall y & \equiv \\
\exists x \forall r \exists y \neg
\end{align}
$$
Now the fun part.  Using De Morgan's law and the fact that $\neg (p \to q) \equiv p \land \neg q$: 
$$
x \in U \land \neg \bigg[ r > 0 \land \big[r \in \Bbb R \to (|y-x|\lt r \to y \in U) \big] \bigg]
$$
$$
x \in U \land \bigg[ \neg (r > 0) \lor \neg \big[r \in \Bbb R \to (|y-x|\lt r \to y \in U) \big] \bigg]
$$
$$
x \in U \land \bigg[ \neg (r > 0) \lor \big[r \in \Bbb R \land \neg (|y-x|\lt r \to y \in U) \big] \bigg]
$$
$$
x \in U \land \bigg[ \neg (r > 0) \lor \big[r \in \Bbb R \land (|y-x|\lt r \land y \notin U) \big] \bigg]
$$
$$
x \in U \land \bigg[  (r > 0) \to \big[r \in \Bbb R \land (|y-x|\lt r \land y \notin U) \big] \bigg]
$$
And finally, if we re-bound the quantifiers, we get
$$
\exists x \in U \forall r > 0 \exists y \in \Bbb{R} : |y-x|\lt r \land y \notin U
$$
So your solution is almost correct, you just got a quantifier wrong.
