Cartesian product of finite set and countable set is countable I want to prove that the product of a finite set and a countable set  is countable. Is it enough to prove that a finite set is countable and use Cantor's theorem to prove that Cartesian product of two countable sets is also countable?
 A: Let the finite set be $\{1,...,N\}$ and let the countable set be $\mathbb{N}$, then define
$\phi(k,n) = k+(n-1)N$. This defines a bijection $\phi:\{1,...,N\} \times \mathbb{N} \to \mathbb{N} $.
A: OK, so your strategy is to use Cantor's proof together with the following Lemma:
Lemma: "Every finite set is countable"
OK, so how do you prove that Lemma?  
First, let's get clear on what we mean by 'countable'. In general, a set $X$ is 'countable' when there is a function from $\mathbb{N}$ to $X$ that is onto (it does not have to be a bijection ... indeed, that would only make infinite sets possibly countable!)
OK, so suppose $X$ is a finite set. Then we can say that $X = \{ x_1,x_2,...x_k \}$ for some $k$
OK, so let's define a mapping $f$ from $\mathbb{N}$ to $X$:
$$f(n) = x_{(n \: mod \: k) + 1}$$
Since this mapping is onto, that means $X$ is countable. So, any finite set is countable.
A: Some people define "countable" to mean countably infinite, but in my experience "countable" usually means that the set can be put in bijection with some subset of $\mathbb{N}$. That is to say, a countable set is either finite or countably infinite. With that definition, a finite set is countable, so Cantor's theorem gives you the result you want.
