finite sets and countable sets.

Question

Let $ A \neq \emptyset $. If $ A $ is finite and $ B $ is countable, then $ A \times B $ is countable.

My attempt is as follows, since $ A $ is finite and $ B $ is countable then $$ A = \{a_1, a_2, \ldots, a_n \} \quad \quad \quad B = \{b_1, b_2, \ldots, b_n ,\ldots \} $$ So $$ A \times B = \{(a_1, b_1), (a_1, b_2), (a_2, b_1), (a_1, b_3), \ldots (a_1, b_n), (a_2, b_ {n-1} ), \ldots, (a_n, b_1), \ldots \} $$ How could I set a bijection from $ \mathbb N \times \mathbb N $ to $ A \times B $, or is there an error in my argument?

Answer 1

The countable union of countable sets is countable, and in your case $$A \times B = \bigcup_{i = 1}^{n} \{a_i\} \times B$$so $A \times B$ is countable.

Answer 2

You need merely a injection from $A\times B$ to $\mathbb N$, so list the elements of $A\times B$ as follows:

$ (a_1,b_1),(a_2,b_1),...,(a_n,b_1),(a_1,b_2),(a_2,b_2),...,(a_n,b_2),...,$

$(a_1,b_m),(a_2,b_m),...,(a_n,b_m),...$