I'm trying to prove that given $n\in \mathbb N$, there isn't a natural number such that $n\lt x\lt n+1$, using the axioms of the natural numbers and the definition of $\lt$ ($m\lt n$ iff $n=m+p$, $p\in \mathbb N$). I've already proved associativity, commutativity and cancellation law of the natural numbers. So we can use this to prove this question, I need help here.
By the way, anyone knows where can I find more exercises like this one in order to train the axioms and first definitions of numbers?
Thanks a lot