Can someone help me finish my solution?
Question: Show that there are sets $A_{ij}$ for $i,j$ ∈ $\mathbb N$ such that for no countable $\space$H$\subseteq\mathbb N^{\mathbb N}$
$\bigcup_{i=0}^\infty\Bigg(\bigcap_{j=0}^\infty{A_{ij}}\Bigg)=\bigcap\Bigg\lbrace\Bigg(\bigcup_{i=0}^\infty A_{ih(i)}\Bigg)\mid h\in H\Bigg\rbrace$..............(1)
Solution:
Assume for all sets $A_{ij}$ , H is countable such that equation (1) holds. As H is countable we can list its elements so we have $\space$
$h_0(0), h_0(1),h_0(2),...\space$
$h_1(0),h_1(1),h_1(2),...\space$
$h_2(0),h_2(1),h_2(2),...$$\space$
.
.
Now I define a function $g(i)$ as such so that it does not appear in the above list. So I go about using Cantor's diagonal argument and define as
$g(i) = \begin{cases} 0 & \text{if } h_i(i) = 1 \\ 1 & \text{if } h_i(i) \neq 1 \end{cases}$
So clearly $g(i)$ is not in the above list. I am now struggling to define $A_{ij}$ so that when I use definition of $g$ on $R.H.S$ I get $L.H.S=R.H.S$. Thus showing that indeed $g(i)$ satisfies equation (1) and yet not listed so it follows that 'there are sets $A_{ij}$ for which H has to be uncountable to satisfy equation (1)'
Can someone help me finish this solution? Thank