showing $\nexists\;\beta\in\mathbb N:\alpha<\beta<\alpha+1$ I want to prove that $\nexists\; \beta\in\mathbb N$ such that $\alpha<\beta<\alpha+1$ for all $\alpha\in\mathbb N$. I just want to use the Peano axioms and $+$ and $\cdot$
If $\alpha<\beta$ then there is a $\gamma\in\mathbb N$ such that $\beta=\alpha+\gamma$.
If $\beta<\alpha+1$ then there is a $\delta\in\mathbb N$ such that $\alpha+1=\beta+\delta$.
Now I tried to equalize the two equations and I got $\gamma\le0$ which is  contradictory to $\gamma\in\mathbb N$. But I used $\alpha+1-\delta=\beta$ in which the $-$ is problematic because I am not allowed to use it.
Anbody knows a better solution? Thanks a lot!

I'm not sure if the following is allowed but:
Inserting the first equation in the second we get:
$\alpha+1=\alpha+\gamma+\delta$.
Now we can substract $\alpha$ at both sides to get $1=\gamma+\delta$, which is a contradiction.
