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 to
. He gives the arrangement as shown in the attached figure and then also gives the formula for that arrangement.
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 to
. 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.