Proof that a sequence of set has a set dense somewhere in \$[a,b]\$ Code Answer

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Is the following proof correct?

Proposition: if we have a sequence of set \$U_i\$ such as \$bigcup_{iin mathbb{N}} U_i=[a,b]\$ then there exist a \$i\$ such as \$U_i\$ is dense somewhere in \$[a,b]\$

Proof: If for all \$i, U_i\$ is nowhere dense, we have \$forall ]u,v[\$

\$\$forall i, exists c in ]u,v[, exists h > 0,quad U_i cap ]c-h,c+h[ =emptysettext{ and } ]c-h,c+h[ subset ]u,v[\$\$

Let’s define

\$\$leftlbrace begin{array} .E_{c,h}^i= ]c-h,c+h[ & text{ if } & U_i cap ]c-h,c+h[ =emptyset \ E_{c,h}^i= emptyset &text{ if } & U_i cap ]c-h,c+h[ neq emptyset end{array}right.\$\$
and
\$\$E_i = bigcup_{h>0} bigcup_{cin ]u,v[} E^i_{c,h}\$\$

\$E_i\$ is open (as a reunion of open set), and dense in \$]u,v[\$. Indeed, we have as

\$\$forall epsilon >0 forall xin ]u,v[, exists c in ]x-epsilon,x+epsilon[,exists h >0,quad U_i cap ]c-h,c+h[ = emptyset\$\$

\$\$forall epsilon >0 forall xin ]u,v[, exists ]c-h,c+h[ subset U_i^C,quad]c-h,c+h[subset ]x-epsilon,x+epsilon[\$\$

\$\$forall epsilon >0 forall xin ]u,v[, exists cin E_i,quad cin ]x-epsilon,x+epsilon[\$\$

So \$E_i\$ is an open dense set for all i. By Baire theorem, we get that \$bigcap_{iinmathbb{N}} E_i\$ is dense in ]u,v[. As we have \$forall i, E_isubset U_i^C\$, we get that \$bigcap_{iinmathbb{N}} E_i subset bigcap_{iinmathbb{N}} B^C_i\$ and \$bigcap_{iinmathbb{N}} B_i^C\$ is dense in \$]u,v[\$. But we also have

\$\$bigcap_{iinmathbb{N}} left(B_i^C cap ]u,v[right) = ]u,v[ cap left( bigcup_{iinmathbb{N}} B_i right)^C = ]u,v[^C cap ]u,v[= emptyset\$\$