Explain a statement in a proof

Question

At a proof at ProofWiki it is proved that there is a neighborhood $U$ of $x_0$ such as $fU\subseteq(c..d)$.

In the proof it is used

$(a): \qquad x \in {U_r}^- \implies f \left({x}\right) \le r$

I haven't understood why this strong statement is required. Why the following is not enough?

$x \in U_r \implies f \left({x}\right) \le r$

Answer 1

It seems that I have got it:

It is necessary to establish that $U = U_q \setminus {U_p}^-$ is an open set and thus a neighborhood of $x_0$. It is not said explicitly in the proof.