Pseudocompact metacompact Tychonoff spaces are compact I was working in some exercises of the famous book Extensions and absolutes of Hausdorff Spaces by Porter and Woods but I'm stuck in some parts of the exercise 6N. The full exercise:

For (1), is clear that $\mathcal{U}'$ is an open cover of $Y$. Suposse that $\mathcal{U}'$ is not point-finite. Take $p\in\beta X\setminus X$ and $\{U_n'\mid n<\omega \}\subseteq \mathcal{U}'$ an infinite collection such that $p\in\bigcap_{n<\omega}U_n'=U$. Then $U$ is a $G_\delta$ set in $\beta X=\upsilon X$ by the pseudocompactness of $X$ and therefore $U\cap X\neq\emptyset$. But then $\bigcap_{n\in\omega} U_n\cap X\neq\emptyset$ and this contradicts the fact that $\mathcal{U}$ is a point-finite famly of $X$.
For (2), is clear that $F_n$ is a cover of $Y$. Take $n\in\mathbb{N}$ and $z\in Y\setminus F_n$. Then $|(\mathcal{U}')_z|>n $ and therefore $(\mathcal{U}')_z=\{U_1',U_2',\dots,U_m' \}$ with $m>n$. Define $V=\bigcap_{i=1}^{m}U_i'$. $V$ is an open set of $Y$ and if $w\in V$ then $|(\mathcal{U}')_w|>n$. Then $z\in V\subseteq Y\setminus F_n$. Therefore $F_n$ is a closed set of $Y$.
For (3). Here, I followed the hint. As $Y$ is an open subset of $\beta X$ and $\beta X$ is compact then $Y$ is locally compact and clearly $T_2$. Then $Y$ is a Baire space and we can conclude that $\bigcup\{\operatorname{int}_{Y}(F_n)\mid n\in\mathbb{N} \}$ is dense in $Y$. How can I conclude from here that $W=\bigcup\{\operatorname{int}_{Y}(F_n)\setminus F_{n-1} \}$ is dense in $Y$? The first idea that came to my mind was to consider an open set $U$ of $Y$ and prove that for all $n\in\mathbb{N}$, $U\setminus F_n\neq\emptyset$. If it is true, $U\setminus F_n$ is an open subset of $Y$ and by the hint we are done. But I think that this is so hard to prove or, actually, is not true. Any hint?
For (4), I followed the hint and made the inductive construction that suggests but in the last part, where I need to prove that $\{V_n\mid n\in\mathbb{N} \}$ is an infinite and locally finite family is where I am stucked. Clearly every $V_n$ is an open subset of $X$ because is the intersection of two open sets of $\beta X$ with $X$. Any idea or suggestion? I tried to prove that if $[\bigcap(U')_{y(n)}]\cap [\bigcap(U')_{y(m)}]\neq\emptyset$ then $y(n)=y(m)$ but I dont know if it is true.
With the another parts of the exercise I'm done and I don't have any trouble. Thanks. I appreciate any help.
Edit. The inductive construction that is suggested in (4):
First, asumme that the union of every finite subfamily of $\mathcal{U}'$ is not dense en $Y$. Take $y(1)\in W$ and consider $\mathcal{U}'_{y(1)}$. This set is a finite subfamly of $\mathcal{U}'$ and therefore there exist $V\subseteq Y$ and open set of $Y$ such that $V\cap \bigcup \mathcal{U}'_{y(1)}=\emptyset$. As $\mathcal{U}'$ is an open cover of $Y$, take $U_2'\in\mathcal{U}'$ such that $U_2'\cap V\neq\emptyset$. Then $U_2'\cap V$ is a non empty open set. Since $W$ is dense (by (3)) then $U_2'\cap V\cap W\neq\emptyset$. Take $y(2)\in U_2'\cap V\cap W$. This $y(2)$ have the desired properties and this completes the basis of induction. Now, suposse that we have been constructed $y(1),\dots,y(n)$ with the desired property. Observe that $\{\bigcup (\mathcal{U}')_{y(k)}\mid k\leq n \}$ is a finite subfamly of $\mathcal{U}'$. Doing the same as in the basis of induction, we can construct a point $y(n+1)$ such that $y(n+1)\in W\setminus \bigcup\{\bigcup (\mathcal{U}')_{y(k)}\mid k\leq n \}$. Now our construction is complete. Let $n\in\mathbb{N}$. As $y(n)\in W$ then there exist $m(n)\in\mathbb{N}$ such that $y(n)\in \operatorname{int}_{Y}(F_{m(n)})\setminus F_{m(n)-1}$.
 A: I agree with $(1)$, where you use that $\beta X = \nu X$ for $X$ pseudocompact and so $X$ is $G_\delta$-dense in $\beta X$ (Thm. 5.11(b) is also used here). So indeed $\mathcal{U}'$ is a point-finite cover of $Y$.
$(2)$: That the $F_n$ are closed is indeed witnessed by the intersection of $\mathcal{U}'_y$ for $y \notin F_n$. Also, the $F_n$ are increasing in $n$ by definition and they cover $Y$ by point-finiteness of $\mathcal{U}'$ (it's good to be explicit).
$(3)$: That $\bigcup_n \operatorname{int}_Y(F_n)$ is dense in $Y$ follows more subtly: suppose not, then some non-empty $Y$-open $O$ misses that union, but then this contradicts the Baire theorem applied to $O$ (which is also locally compact Hausdorff, hence a Baire space), as the $F_n \cap O$ then are nowhere dense in $O$ but cover $O$, making $O$ first category in itself.
So given a non-empty $O$ open in $Y$, we know $O$ intersects $\bigcup_n \operatorname{int}_Y(F_n)$ so there is minimal $n_0$ so that $O \cap \operatorname{int}_Y(F_n) \neq \emptyset$ and then $O$ also intersects $\operatorname{int}_Y(F_{n_0}) \setminus F_{n_0-1}$ by that minimality, proving the claim that $W$ is dense. Openness of $W$ is obvious.
The recursive construction  of the $y(n)$ you gave is fine IMO, so the only question is local finiteness of the $(V_n)$, $n \in \Bbb N$.
I'm still thinking on $(4)$, but I made it a Community wiki so others (Brian maybe?) can add to and improve the answer.
