Statement: if m and n are natural numbers this condition can't happen: n<m<n+1

Proof: So I said that let's assume that n<m<n+1 is true.

So from the first part we get n<m and from here we get that m-n is a natural number.

So there exist k in N and we can express m-n=k+1 (I depended on a statement that I proved before)

and from here we get m-k=n+1

So from the second part of the inequality we get m<n+1=m-k and from here we get that k<0 and this a contradiction to the fact that we can express m-n as k+1 and therefore this condition can't happen.

Happy if someone can give feedback thanks!