Here, page $457$, in the proof of theorem $1$, there is this sentence
'Select a nested sequence $\{ U(n,i,j)| j=1,2,... \}$, of open subsets of $I$ whose intersection is $f^{-1}[\frac{i}{n}, \frac{i+1}{n}].$'
I understand where the sequence of open subsets come from (by the definition of Baire -$1$), but I don't understand how can we select a nested sequence.
Can anyone help?
You know that there is a sequence $\langle V_j:j\in\Bbb Z^+\rangle$ of open sets whose intersection is
$$f^{-1}\big[[i/n,(i+1)/n]\big]\;.$$
For $j\in\Bbb Z^+$ let $$U(n,i,j)=\bigcap_{i=1}^jV_i\;;$$
then $U(n,i,j)\supseteq U(n,i,j+1)$ for each $j\in\Bbb Z^+$, and
$$\bigcap_{j\ge 1}U(n,i,j)=\bigcap_{j\ge 1}V_j=f^{-1}\big[[i/n,(i+1)/n]\big]\;,$$
as desired.
This is a standard trick: any sequence of open sets with the finite intersection property can be converted to a nested sequence of open sets with the same intersection by progressively intersecting more and more of the sets in the original sequence.