How to prove this claim on $\omega$ How to prove this claim: $T\subset \omega$ and if $n \subset T$ implies $n\in T$, then $T=\omega$?
Any help will be appreciated.

A hint: Argue by induction, or what is the same, show that there can be no smallest witness to the negation of the conclusion.
In more detail: If $T\ne\omega$ then, since $T\subset\omega$, there must be some $n\in\omega$ with $n\notin T$. Then the set of elements of $\omega$ that are not in $T$ is nonempty, so it has a smallest element, say $m$. Then $m\notin T$ but any $k<m$ is in $T$. It should be clear how to finish from here.
