We write $A\subseteq^\* B$ if $A\setminus B$ is finite.


Let $(A\_n)\_{n\in\omega}$ be a sequence of subsets of $\omega$ such that for all $n\in\omega$ we have $A\_n \subseteq^\* A\_{n+1}$ and $A\_{n+1}\not\subseteq^\* A\_n$.


Let $D\_n:= A\_{n+1}\setminus A\_n$ for all $n\in\omega$. Is it possible that $$\bigcup\_{n\in\omega} D\_n = \omega?$$



Can the union of difference sets in towers equal $\omega$?
Take any increasing tower, but then modify it by adding all the numbers below $n$ to $A\_n$, when $n$ is even, and removing them when $n$ is odd. This is a finite change to each set in the tower, and so it doesn't affect any $\subseteq^\*$ relation, but now every number will eventually appear in the difference sets, jumping in and then out again at every other step. 


