Let $X$ be a topological space. The point-open game $G_{po}(X)$ is defined as folows. It is played by two players ONE and TWO. In the n'th step $(n \in \omega)$, ONE choose a finite subset $F$ of $X$, and TWO selects an open $G_n$ in $X$, $F_n \subset G_n$. ONE wins if $\bigcup \{ G_n : n \in \omega \} = X$, otherwise TWO wins.
Also:
If $\langle A_n : n \in \omega \rangle$ is a sequence of subsets of a set $X$, $$ \underline{Lim} A_n = \{ x \in X : \exists n_0 \in \omega \forall n \geq n_0, x \in A_n \} $$ If $\mathcal A$ is a family of subsets of a set $X$, then, $L(\mathcal A)$ denotes the smallest family of subsets of $X$ containing $\mathcal A$ and closed under $\underline{Lim}$.
I am trying to prove that (a)$\Rightarrow$(b) where:
(a) If $\mathcal I$ is an open $\omega$-cover of $X$, then, there is a sequence $G_n \in \mathcal I$, with $\underline{Lim} G_n = X$.
(b) If $\mathcal I$ is an open $\omega$-cover of $X$, then $X \in L(\mathcal I)$.
A family $\mathcal A$ of subsets of a set $A$ is said to be an $\omega$-cover of $X$, if for any finite subset $F$ of $X$,, there is an $A \in \mathcal A$ with $F \subset A$.