Hello

I have some questions about proofs presented while proving that some set is

countable. In Velleman's "How to prove it" , author is trying to prove that there is

a bijection from $\displaystyle \mathbb{Z^+}\times \mathbb{Z^+}$ to

$\displaystyle \mathbb{Z^+}$. He gives the arrangement as shown in the attached figure and then also gives the formula for that arrangement.

$\displaystyle f(i,j)=\frac{(i+j-2)(i+j-1)}{2}+i$

and then he asks the reader to show that the above function is indeed a bijection

between the said sets. But I have seen the arguments where people just give the

triangular arrangement as shown in the figure and argue that the arrangement is

one to one and onto between $\displaystyle \mathbb{Z^+}\times \mathbb{Z^+}$ to

$\displaystyle \mathbb{Z^+}$. The wikipedia article on "countable set" has such arguments. My question is , are these valid arguments ? Velleman's way of reasoning seems more rigorous to me. We need to come up with some functional

form of the given arrangement and then prove using usual methods that its a

bijection. Please comment and move this thread to sub forum where its appropriate.

Thanks