Proving that $A_1\times A_2 \times \cdots \times A_n$ is countable If $A_1$,$A_2$,...,$A_n$ are countably infinite sets, is $A_1\times A_2\times...\times A_n$ countable?
it seems that an infinite Cartesian product of countable sets may still be uncountable for every x∈[0,1) in a countable set.

Let $A_i=\{ a_{i,j} \}_{j\in\mathbb{N}}$ for $i=1,\dots,n$. Let $p_1,\dots,p_n$ the first $n$ primes.
The function $\varphi: A_1\times\dots\times A_n\rightarrow\mathbb{N}$ defined by
$$
\varphi(a_{1,j_1},\dots,a_{n,j_n}):=p_1^{j_1}\cdots p_n^{j_n}
$$
is an injection. This shows that $|A_1\times\dots\times A_n|\le \aleph_0=|\mathbb{N}|$. On the other hand, $A_1\times\dots\times A_n$ is an infinite set, so
$$
|A_1\times\dots A_n|=\aleph_0=|\mathbb{N}|.
$$
