The proof looks OK but is there a specific reason why you wanted to show it this way as opposed to another way?
Thought of this today.
two years ago someone told me that you cannot know if there exist a biggest integer or not.
then i said that there must be a bigger integer, because we can add one to it.
But now i thought , so when we get to big numbers adding one does not increase it.
keep in mind i have a very elementary knowedge of infinity, so maby this is supid i dont know.
In this proof i will assume that:
1. No integer can be both odd and even at the same time.
2. For any integer n,
1 and 2 are not proven here, but proof exist.
assertion: there exist many integers.
proof: suppose n is the biggest integer such that But if n is even, then from def for an integer k.
But then for some k, which is odd by def. Then we know that because of assumption 1. And we know that by assumption 2.
Then the only option that remains is But then n is not the biggest integer.
But if n is odd then from def for some k, But then which is even an even integer from def. Thus by assumptions 1,2 the only option is
therefore if n is odd or even, there exist a bigger integer n+1. Q.E.D.
please point out any flaws in my arguments you may find.
no not really, i am still a novice on proofs, so i thought that this would be simple and fun to try to prove.
i am sure that there exist many better and simpler proofs of this fact, but this is what i came up with.