Prove that the set of triples $\{(a,b,c)|a,b,c \in \mathbb{N}\}$ is countable I have the following question in my textbook:
Prove that the set of triples $\{(a,b,c)|a,b,c \in \mathbb{N}\}$ is countable
Now I know that $\mathbb{N}$ is countable already, and I have completed a non-rigorous proof of this before, but I am unsure of how having a set of triples changes things, nor do I understand what a set of triples pertains to. I assume I am missing some fundamental knowledge in solving this problem?
My guess is that a set of triples is: $(a,b,c)$ so $(1,1,1),(1,1,2),(1,1,3)...(2,1,1),(2,1,2)...(3,1,1)...$ etc
I imagine I can show this to be countable via pictures if my guess is correct. But how would I go about proving it rigorously?

It is possible to show this without actually constructing a bijection. Let 
$$S=\{(a,b,c)|a,b,c\in \mathbb N\}.$$
If you define $$A_n=\{(a,b,c)\in S|a+b+c=n\},$$
then it is clear that $A_n$ is finite for every value $n\in \mathbb N$. For example, $|A_1|=|A_2|=0$, $|A_3|=1$, $|A_4|=3$ and so on.
Then, you can show that $$S=\bigcup_{i\in\mathbb N} A_i.$$
Because a countable union of finite set is countable, that means that $S$ is countable.
