In the Engelkings book "General Topology" in the chapter of paracompactness there is a lemma related to partitions of unity and locally finite covers.
I am trying to understand the proof of the lemma, but I am blocked at the part when $V_s$ sets are defined. I can't get why $V_s$ are open, why $\mathcal{V}$ is a refinement of $\mathcal{U}$ and why $\mathcal{V}$ is locally finite.
The lemma together with the proof is given below:
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For $s\in S$ let $h_s(x)=f_s(x)-\frac12f(x)$; $h_s$ is continuous (because $f_s$ and $f$ are), so
$$V_s=\{x\in X:h_s(x)>0\}=h_s^{-1}\big[(0,\to)\big]$$
is open. For each $x\in V_s$ we have $\color{red}{f_s(x)>\frac12f(x)}>0$ by the definition of $f$, so $x\in f_s^{-1}\big[(0,1]\big]$. The partition of unity is subordinated to $\mathscr{U}$, so by definition there is a $U\in\mathscr{U}$ such that $V_s\subseteq f_s^{-1}\big[(0,1]\big]\subseteq U$, and $\mathscr{V}$ is a refinement of $\mathscr{U}$.
Finally, let $g=\frac12f$, and let $x\in X$; $f(x)>0$, so $g(x)>0$, and therefore there are a nbhd $U_0$ of $x$ and a finite $S_0\subseteq S$ such that $f_s(y)<g(y)=\frac12f(y)$ for all $y\in U_0$ and $s\in S\setminus S_0$. Let $s\in S\setminus S_0$ and $y\in U_0$; then $f_s(y)<\frac12f(y)$, so $y\notin V_s$ (by the red inequality above). Thus, $U_0\cap V_s=\varnothing$ for all $s\in S\setminus S_0$, so $\{s\in S:U_0\cap V_s\ne\varnothing\}\subseteq S_0$, and $U_0$ is therefore a nbhd of $x$ that meets only finitely many members of $\mathscr{V}$. Since $x\in X$ was arbitrary, this shows that $\mathscr{V}$ is locally finite.