We are now ready for the proof of Theorem \[theorem-Near-sel-v1:1\].

Let $\varphi$ and $\varepsilon$ be as in this theorem. Since $X$ is a paracompact space, by Proposition \[proposition-Near-sel-v1:1\], it has an open locally finite cover $\mathscr{V}$ and a function $\delta:\mathscr{V}\to (0,+\infty)$ such that $$\label{eq:Near-sel-v1:1}
    \delta(V)\leq \varepsilon(p),\quad \text{for every
      $p\in V\in \mathscr{V}$.}$$ Hence, by Proposition \[proposition-Finite-C-v4:3\] (applied with $\mathscr{U}_1=\mathscr{V}$ and $\delta_1=\delta$), $X$ has an open locally finite cover $\mathscr{U}_0$ which refines $\mathscr{V}$, and maps $\delta_0,\varepsilon_0: \mathscr{U}_0\to (0,+\infty)$ with $\delta_0\leq \varepsilon_0$, such that for every $p,q\in U\in \mathscr{U}_0$, $$\label{eq:Near-sel-v1:2}
    \begin{cases}
      \mathbf{O}_{\delta_0(U)}(\varphi(p)){\lhook\joinrel\xrightarrow{*}}
      \mathbf{O}_{\varepsilon_0(U)}(\varphi(p)),\\[3pt]
      H(\rho)(\varphi(p),\varphi(q))<\frac{\delta_0(U)}2,\\
      \varepsilon_0(U)\leq \frac{\varepsilon(p)}2.\\
    \end{cases}$$ To see the last property in , take $p\in U\in \mathscr{U}_0$ and $V\in \mathscr{V}$ with $U\subset
  V$. Then by and Proposition \[proposition-Finite-C-v4:3\], $\varepsilon_0(U)\leq \frac{\delta(V)}3\leq
  \frac{\varepsilon(p)}3\leq \frac{\varepsilon(p)}2$.

Using Proposition \[proposition-Finite-C-v4:3\], the construction can be carried on by induction. Thus, there exists a sequence $\mathscr{U}_n$, $n<\omega$, of locally finite open covers of $X$ and functions $\delta_n,\varepsilon_n:\mathscr{U}_n\to (0,+\infty)$ with $\delta_n\leq \varepsilon_n$, such that each $\mathscr{U}_{n+1}$ refines $\mathscr{U}_n$ and the following properties hold for each $p,q\in U\in \mathscr{U}_{n}$: $$\label{eq:Near-sel-v2:1}
    \begin{cases}
      \mathbf{O}_{\delta_n(U)}(\varphi(p)){\lhook\joinrel\xrightarrow{*}}
      \mathbf{O}_{\varepsilon_n(U)}(\varphi(p)),\\[3pt]
      H(\rho)(\varphi(p),\varphi(q))<\frac{\delta_n(U)}2,\\
      \varepsilon_{n+1}(V)\leq \frac{\delta_n(U)}3,\ \text{whenever
        $V\in \mathscr{U}_{n+1}$ with $V\cap U\neq {\varnothing}$.}
    \end{cases}$$

To finish the proof, for each $n<\omega$ take a map $\pi_n:\mathscr{U}_n\to X$ with $\pi_n(U)\in U$, $U\in \mathscr{U}_n$. 