Let $A$ be an countable set, and let $B_n$ be the set of all $n$-tuples $\left(a_1,\ldots,a_n\right)$
$B_n$ is the union of a countable set of countable sets.
This question maybe about the English.
Is my rephrase right?
$B_n$ is a countable set as the union of countable sets.
I think the quoted sentece has some problems in grammar or others?
Of course, that's only my judge, I'm not a native speaker and not so confident in the judge, so I asked this.