A meagre set is always contained in an $F_σ$ set made from nowhere dense sets. In this page I have found a beautiful result that a meagre set need not be an $F_σ$ set (countable union of closed sets), but is always contained in an $F_σ$ set made from nowhere dense sets.
Also dually, just as the complement of a nowhere dense set need not be open, but has a dense interior (contains a dense open set), a comeagre set need not be a $G_δ$ set (countable intersection of open sets), but contains a dense $G_δ$ set formed from dense open sets.
I am able to prove the dual result but unable to prove the first one. I think we can use the dual result to prove the first one.
Please Help!!

Suppose $A$ is a meagre set, i.e. it is the union of countably many nowhere dense sets $B_j$.  "Nowhere dense" means that the closures $\overline{B_j}$ contain no nonempty open set.  These closures are also nowhere dense.  Then $A$ is contained in $\bigcup_j \overline{B_j}$, which is an $F_\sigma$ made from nowhere dense sets.
