No, only if $\omega$ is a p-point.
If $(A_k:k\in \mathbb N)$ is a partition of the natural numbers into $\omega$-small sets such that there is no $\omega$-large set meeting each $A_n$ A_k$in a finite set, then we can choose a sequence$x_n$by declaring$x_n:= 1/k$whenever$n\in A_k$. On each set$J\in \omega$the sequence$x_n$does not converge to 0. To find a non-p-point, partition$\mathbb N$into countably many infinite sets$A_k$, let$I$be the ideal of sets meeting each$A_k$only finitely often, and let$\omega$be dual to any maximal ideal extending$I$. Note that ZFC does not prove that there are p-points, but existence of p-points follows from CH (or weaker assumptions). 1 No, only if$\omega$is a p-point. If$(A_k:k\in N)$is a partition of the natural numbers into$\omega$-small sets such that there is no$\omega$-large set meeting each$A_n$in a finite set, then we can choose a sequence$x_n$by declaring$x_n:= 1/k$whenever$n\in A_k$. On each set$J\in \omega$the sequence$x_n\$ does not converge to 0.