Hi!

Problem:

Prove that the sets $\displaystyle \mathbb{N} \times \mathbb{N} = \left\{ (i,j): i,j \in \mathbb{N} \right\} $ and $\displaystyle \mathbb{Q} $ are denumerable.

(If someone wants to give a precise definition of denumerable please do).

For a fixed $\displaystyle j \in \mathbb{N} $ the set $\displaystyle \left\{ (i,j): i,j \in \mathbb{N} \right\} $ is denumerable since $\displaystyle \mathbb{N} $ is denumerable.

If we change $\displaystyle j $ to $\displaystyle j' $, the set $\displaystyle \left\{ (i,j'): i,j' \in \mathbb{N} \right\} $ is also denumerable.

So we have a denumerable amount of denumerable sets, which gives that $\displaystyle \mathbb{N} \times \mathbb{N} $ is denumerable?

I tried proving $\displaystyle \mathbb{Q} $ in a similar way.

Help appreciated!