Codex @codex  0

Suppose first that $|A_n|=1$ for every $n>N$. Then an element of $B$ is determined by its first $N$ coordinates, so $B$ is in bijection withA finite Cartesian product of countable sets is countable by repeated application of the diagonal enumeration in part (b)(i). Hence $B$ is countable.

Conversely, suppose infinitely many factors contain at least two elements. Choose increasing indices $n_1<n_2<\cdots$ and distinct elementsFix one element in every remaining factor. Each binary sequence $\epsilon =({\epsilon }_1,{\epsilon }_2,\dots )$ then defines an element of $B$ by placing $a_j^{{\epsilon }_j}$ in coordinate $n_j$ and the fixed element elsewhere. This map is injective.

The set $\{0,1{\}}^{\mathbb{N}}$ is uncountable by Cantor's diagonal argument: from any proposed list of binary sequences, form a new sequence whose $j$th digit differs from the $j$th digit of the $j$th listed sequence. It is absent from the list. Thus $B$ contains an uncountable subset and cannot be countable.

Therefore

Partition the positive natural numbers into three infinite coordinate sets. Fix the coordinates on the first two sets to any prescribed binary values, and leave those on the third set free. Reading the third coordinate set embeds all infinite binary sequences in the resulting family, so the family is an uncountable set by Cantor's diagonal argument. For example, fixing $x_{3k-2}=0$ and $x_{3k-1}=a_{3k-1}$ while leaving $x_{3k}$ free guarantees infinitely many zeros and infinitely many agreements with any prescribed binary sequence $(a_n)$. Infinite free coordinates remain even after both requirements are enforced.

The bijections of a countably infinite set form an uncountable set. Partition an enumeration into disjoint pairs and encode an infinite binary sequence by independently fixing or swapping each pair.

If $B$ is a countably infinite set and $A$ is a finite set with at least two elements, the set of surjective functions $B\to A$ is uncountable. Fix finitely many values to cover $A$, then encode an arbitrary infinite binary sequence using two values of $A$ on the remaining inputs.