Well-known the definition of first selection principle $S_1(A,B)$. Let $A$ and $B$ be families of sets. The symbol $S_1(A, B)$ denotes the statement: for each sequence $(A_n : n \in \omega)$ of elements of $A$ there is a sequence $(x_n : n\in \omega)$ such that each $x_n$ is an element of $A_n$, and $\{x_n : n\in \omega\}$ is an element of $B$.

Can we assume that $x_{n_k}=\emptyset$ for $k\in \omega$ ?