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9 Remove ambiguities from the proof, use a more convenient metric; [made Community Wiki]

Thanks for all the answers and sorry about a silly question. I have also figured out that it can be proved using the usual complete metric on the usual (countable product) Hilbert cube and finite $\epsilon$-nets.

Update. Here is a "meta-proof" (i do not construct $\epsilon$-nets in details)proof.

Let $X = [0,1]\times[0,\frac{1}{2}]\times[0,\frac{1}{3}]\times\dotsb$ -\frac{1}{2},\frac{1}{2}]\times[-\frac{1}{4},\frac{1}{4}]\times[-\frac{1}{8},\frac{1}{8}]\times\dotsb$be a Hilbert cube endowed with its$\ell_2$\ell_\infty$ metric. For every positive integer $k$, let $N_k$ be a "well chosen" the natural” $\frac{1}{2^k}$-net for $X$.

Let $\mathcal U$ be a given family of open sets such that no finite subfamily of $\mathcal U$ covers $X$. Then let $S_k$ be the set of those elements of $N_k$ which are within the distance of $\frac{1}{2^k}$ from the complement of any finite union of elements of $\mathcal U$. Each $S_k$ is nonempty. For all $m$ and $n$, the distance from any point of $S_m$ to the set $S_n$ is at most $\frac{1}{2^m}+\frac{1}{2^n}$.

Assume that the points of $X$ are ordered by the lexicographic order of their coordinates. Take "the first" first” $x_1\in S_1$ (i.e. the smallest in the order), then "the first" first” $x_2\in S_2$ that is within the distance of $\frac{3}{4}$ from $x_1$, then "the first" first” $x_3\in S_3$ that is within the distance of $\frac{3}{8}$ from $x_2$, and so forth. The obtained sequence $\lbrace x_k \rbrace$ rbrace_{k=1}^\infty$is Cauchy. Its limit is not in any element of$\mathcal U$. 8 "compliment" -> "complement" Thanks for all the answers and sorry about a silly question. I have also figured out that it can be proved using the usual complete metric on the usual (countable product) Hilbert cube and finite$\epsilon$-nets. Update. Here is a "meta-proof" (i do not construct$\epsilon$-nets in details). Let$X = [0,1]\times[0,\frac{1}{2}]\times[0,\frac{1}{3}]\times\dotsb$be a Hilbert cube endowed with its$\ell_2$metric. For every$k$, let$N_k$be a "well chosen"$\frac{1}{2^k}$-net for$X$. Let$\mathcal U$be a given family of open sets such that no finite subfamily of$\mathcal U$covers$X$. Then let$S_k$be the set of those elements of$N_k$which are within the distance of$\frac{1}{2^k}$from the compliment complement of any finite union of elements of$\mathcal U$. Each$S_k$is nonempty. For all$m$and$n$, the distance from any point of$S_m$to the set$S_n$is at most$\frac{1}{2^m}+\frac{1}{2^n}$. Take "the first"$x_1\in S_1$, then "the first"$x_2\in S_2$that is within the distance of$\frac{3}{4}$from$x_1$, then "the first"$x_3\in S_3$that is within the distance of$\frac{3}{8}$from$x_2$, and so forth. The obtained sequence$\lbrace x_k \rbrace$is Cauchy. Its limit is not in any element of$\mathcal U$. 7 fix "intersection" -> "union" Thanks for all the answers and sorry about a silly question. I have also figured out that it can be proved using the usual complete metric on the usual (countable product) Hilbert cube and finite$\epsilon$-nets. Update. Here is my attempt to fix this proof, please tell me if i have missed something. This is a "meta-proof" (i do not construct$\epsilon$-nets in details). Let$X = [0,1]\times[0,\frac{1}{2}]\times[0,\frac{1}{3}]\times\dotsb$be a Hilbert cube endowed with its$\ell_2$metric. For every$k$, let$N_k$be a "well chosen"$\frac{1}{2^k}$-net for$X = [0,1]^\omega$. Suppose X$.

Let $\mathcal U$ is an be a given family of open cover sets such that no finite subfamily of $X$ without a finite subcover. Let \mathcal U$covers$X$. Then let$S_k$be the set of those elements of$N_k$which are within the distance of$\frac{1}{2^k}$from the compliment of any finite intersection union of elements of$\mathcal U$. Each$S_k$is nonempty. For all$m$and$n$, the distance between from any point of$S_m$and to the set$S_n$is at most$\frac{1}{2^m}+\frac{1}{2^n}$. Take "the first"$x_1\in S_1$, then "the first"$x_2\in S_2$that is within the distance of$\frac{3}{4}$from$x_1$, then "the first"$x_3\in S_3$that is within the distance of$\frac{3}{8}$from$x_2$, etcand so forth. The obtained sequence$\lbrace x_k \rbrace$is Cauchy, and its . Its limit is not in any element of$\mathcal U\$.

6 fix math braces
5 Format and be more explicit
4 take "the first" x_1,...
3 Give proof details.
2 update: the argument does not seem to work
1