Baire category theorem to show a set is dense.

Question

Consider $A_j$ a sequence of subsets of $[0,1]$ s.t. for each $N\geq 1$, $\bigcup_{j=N}^\infty A_j$ is open and dense in $[0,1]$.

If $S$ is the set of points $x \in [0,1]$ s.t. $x \in A_j$ for infinitely many $j$, then I wish to show that $S$ is dense. I am also unsure if $S$ needs to be an open set.

I have tried to find a way to apply the Baire category theory, but I haven't had any success. Thank you for your help.

Answer 1

If $S=\{x\in[0,1]\,|x\in A_j\,,j=1,2,...\}$ then $S=\{x\,|\,x\in\cap_{j=1}^\infty A_j\}$, since all $A_j$ lay in $[0,1]$, we can see that:

$S=\cap_{j=1}^\infty A_j$, and by baire category theorem, this set is dense.

Also I believe $S$ may not need to be open.

I did not take in the first part where $\cup_{j=N}^\infty A_j$ is open and dense in $[0,1]$ $\forall N\ge 1$, but this implies that each $A_j$ is open and dense in $[0,1]$, so the above holds.