a fun proof of inf integers, tell me what you think.

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 $\infty+1 = \infty$ , so when we get to $\infty$ 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, $n+1\not< n.$

1 and 2 are not proven here, but proof exist.

assertion: there exist $\infty$ many integers.
proof: suppose n is the biggest integer such that $n = n+1.$ But if n is even, then from def $n = 2k,$ for an integer k.

But then $n+1 = 2k+1,$ for some k, which is odd by def. Then we know that $n\neq n+1,$ because of assumption 1. And we know that $n+1 \not< n$ by assumption 2.
Then the only option that remains is $n+1 > n,$ But then n is not the biggest integer.

But if n is odd then from def $n = 2k+1,$ for some k, But then $n+1 = 2k+2 = 2(k+1),$ which is even an even integer from def. Thus by assumptions 1,2 the only option is $n+1 > n.$
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.
thanks :)

Re: a fun proof of inf integers, tell me what you think.

Hey engpro.

The proof looks OK but is there a specific reason why you wanted to show it this way as opposed to another way?

Re: a fun proof of inf integers, tell me what you think.

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. :)