I suppose I'd do proof by contradiction but this is one of those proofs that seems so obvious that I have a hard time knowing where to start. I know it's a countably infinite set, so how can I formally show that there cannot be an upper bound?
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I don't know how your class formally defined the integers or bounded from above. here is how I would do it:
Suppose Z+ has an upper bound, n.
Then by definition, there is no member of Z q such that q>n.
But by definition of Z+, if z belongs to z then the successor of z belongs to Z+ (or "z + 1 belongs to Z+").
So n has a successor (or "n+1 belongs to Z+).
But by definition of ">" the successor of n is greater than n, so we have a contradiction.
Suppose Z+ has an upper bound, n.
Then by definition, there is no member of Z q such that q>n.
But by definition of Z+, if z belongs to z then the successor of z belongs to Z+ (or "z + 1 belongs to Z+").
So n has a successor (or "n+1 belongs to Z+).
But by definition of ">" the successor of n is greater than n, so we have a contradiction.