# Banach Mazur Game: Oxtoby Measure and Category

I have a question regarding the proof of theorem 6.2 which states that,

Thm 6.1: There is a strategy in which is sure to win iff is of first category

The game played is this: there is a set $B \subseteq I_0$, where $I_0$ is a closed interval (all intervals here are required to have positive length). The first player $(A)$ chooses a closed interval $I_1 \subseteq I_0$, and then they alternate forever, choosing a closed subinterval $I_{k+1}$ of the previous interval $I_k$. Player (A) wins if the intersection $\bigcap_k I_k$ is disjoint from $B$, otherwise player (B) wins.

The proof is given here below and my question is regarding the second line:

How can we know that given $f_1$, it is possible to define a sequence of closed intervals $J_i$ contained in $I_0^o$ such that the intervals $K_i$ are disjoint? and also, how do we make sure that the union of their interior is dense??

Thank you, Shir