I am reading Demailly's notes on pseudodifferential operators on manifolds. And I cannot understand a statement he had made when he tried to prove that the image of an elliptic differential operator is closed. [The notes is available here](https://www-fourier.ujf-grenoble.fr/%7Edemailly/analytic_geometry_2019/pseudodiff.pdf). And you can find the following proof on page 14.


Here is part of his original proof:
He claims that for any $\epsilon>0$, there exists a finite number of elements $v\_{1},...,v\_{N}\in W^{s+\delta}(M)$, $N=N(\epsilon)$, such that $$
||u||\_{0}\leq\epsilon||u||\_{s+\delta}+\sum\_{j=1}^{N}|<u,v\_{j}>\_{0}|
$$
Here $W^{i}$s are the Sobolev spaces of order $i$ whoes norms are subscribed with the corresponding index $i$. And $<,>$ is the inner product in the Hilbert space $W^{0}=L^{2}$.  

To prove the statement, Demailly constructed a family of sets:$$
K\_{(v\_{j})}=\{u\in W^{s+\delta};\epsilon||u||\_{s+\delta}+\sum\_{j=1}^{N}|<u,v\_{j}>\_{0}|\leq1\}
$$
that are relatively compact in $W^{0}$ (this is true by Rellich's embedding lemma)and $\bigcap\_{(v\_{j})} K\_{(v\_{j})}={0}$(if the intersection was taken among arbitrary indexes ($v\_{i}$), then it is clearly true). Then he concluded that:  

there is a finite sequence $(v\_{j})$ such that $K\_{(v\_{j})}$ is contained in the unit ball of $W^{0}$.


This statement is what I am confused with. Indeed I have proved that if for an increasing sequence of $(v\_{j})$ we have $\bigcap\_{(v\_{j})}K\_{(v\_{j})}=\{0\}$, then his statement is true. But the main issue here is that (in my opinion)the intersection should be taken among uncountably many indexes $(v\_{j})$ to garantee it shrinking to $\{0\}$.



A problem on Demailly's proof of finiteness theorem of elliptic differential operator
$\def\eps{\varepsilon}\def\rbbR{\mathbb R^+}\def\bbNo{\mathbb N\_0}\def\r#1{{\rm#1}}\def\seq#1{\langle\,#1\,\rangle}$It seems to me that Demailly's unhappy vague notation is the main cause of confusion. Actually, the argument in this part of the proof goes e.g. as follows. Using separability of $F=L^2$ and density of $E=W^{s+\delta,2}$ in $F$ and letting $\boldsymbol v=\seq{v\_j:j\in\bbNo}$ be any $F$−dense sequence in $E$, for any $\eps\in\rbbR$ letting $\r N\,(\eps,i,u)=\eps\,\|u\|\_{s+\delta,2}+\sum\_{j=0}^i|(u,v\_j)\_0|$ and $\r B\,i=\{\,u:\r N\,(\eps,i,u)\le 1\,\}$ and $\r K\,i={\rm Cl\,}\_F\,\r B\,i$, by Rellich–Kondrachov then $\seq{\r K\,i:i\in\bbNo}$ is a nonincreasing sequence of compact sets in $F$. One first shows that $\bigcap\,\{\,\r K\,i:i\in\bbNo\,\}=\{\,0\_F\}$ holds whence letting $B$ be the open unit ball in $F$ and $\r U\,i$ the complement of $\r K\,i$ we have $\r K\,0\subseteq B\cup\bigcup\{\,\r U\,i:i\in\bbNo\,\}$. Then by compactness there is $i\in\bbNo$ with $\r K\,i\subseteq\r K\,0\subseteq B\cup\r U\,i$ and hence $\r K\,i\subseteq B$ whence the final assertion.


The proof of $\bigcap\,\{\,\r K\,i:i\in\bbNo\,\}\subseteq\{\,0\_F\}$ goes as follows. Supposing it fails, there is a nonzero $u$ in the intersection of all $\r K\,i$. Since $\boldsymbol v$ is dense in $F$, it has a subsequence converging in $F$ to $u$ whence we see that there is $i\in\bbNo$ such that $1<\sum\_{j=0}^i|(u,v\_j)\_0|$ holds. Now there some $u\_9\in\r B\,i$ with $\r N\,(\eps,i,u\_9)\le 1<\sum\_{j=0}^i|(u\_9,v\_j)\_0|\le\r N\,(\eps,i,u\_9)$, a *contradiction*.


