= Solution
The sheaf $A_n$ is noninitial: it contains $y_n$, whose value at $n$ includes $1_n$. In a <two-valued topos> its support, the image of $A_n\to1$, can therefore only be $1$. An image equal to $0$ would force $A_n=0$, since the <initial object> of a topos is strict. Consequently \b[every $A_n\to1$ is an <epimorphism>].
Products of sheaves on a small site exist and are computed as products of the underlying presheaves, since a product of unique matching-family amalgamations again uniquely amalgamates. For the countable product $P=\prod_{n\in\mathbb N}A_n$, its value at stage $m$ is $\prod_nA_n(m)$. The factor with $n=m+1$ is empty by the stronger statement proved in the preceding part. Thus $P(m)=\varnothing$ at every stage $m$, with the unique empty restriction maps. It is the empty presheaf, already an initial sheaf because no empty <sieve on a category> covers. Hence
$$
\boxed{A_n\twoheadrightarrow1\text{ for every }n,\qquad\prod_{n\in\mathbb N}A_n\cong0.}
$$
This proves the final PDF-only request and exhibits the <separating tower site with vanishing product>. It is a genuine failure of countable choice in this Boolean two-valued <sheaf topos>; it does not contradict the classical external construction of the site.
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