I have to prove that if $n\in \mathbb{N}$, there is no natural number between $n<x<n+1$. I started proving it using Induction.
Base case for $n=1$:
... then $1<x<2$. Now I need a valid justification that this is a true statement in $\mathbb{N}$. What would be a good argument? I guess since $\mathbb{N}$ is inductive and therefore $1\in \mathbb{N}$ and if $n\in \mathbb{N} \Rightarrow n+1\in \mathbb{N}$, hence there is no natural number between $1$ and $2$.
Is this correct?
If $1<x<2$, then $x-1<2-1=1$. This is impossible, since $x-1\in\mathbb N$ and no natural number is smaller than one.
By the way, the same idea can be used to prove what you want to prove without induction.