The union of a countable set of countable sets? 
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.
 A: Here are the two sentences in question:


*

*$B_n$ is the union of a countable set of countable sets.  

*$B_n$ is a countable set as the union of countable sets.


Sentence (1) is just fine, and sentence (2) is not a correct paraphrase of sentence (1).


*

*Sentence (2) says that $B_n$ is a countable set; sentence (1) does not say this.  

*Sentence (1) says that $B_n$ is the union of countably many countable sets; sentence (2) merely says that $B_n$ is the union of some arbitrary family of countable sets, not necessarily a countable family.

A: Let me rephrase the question. Let $A$ be a countable set and let $B_n$ be the set of all $n$-tuples of elements of $A$. 
For a fixed $n$ the set $B_n$ is countable by a diagonal argument. 
I suspect the poser really wants to know about the set $B$ which is the union of all of the $B_n$. This set $B$ is also countable. To phrase this in terms of cardinal arithmetic, 
$$
\begin{align}
& \operatorname{Card}(B_1) + \operatorname{Card}(B_2) + \operatorname{Card}(B_3) + \cdots \\[10pt]
& = \aleph_0 + \aleph_0 \cdot \aleph_0 + \aleph_0 \cdot \aleph_0 \cdot \aleph_0 + \cdots \\[10pt]
& = \aleph_0
\end{align}
$$
