Question on inverse limits 
1.7. Remark. The inverse limit of an inverse system of non-empty sets might be empty as the following example
  shows: Let $I:=\mathbb{N}$ and $X_n:=\mathbb{N}$ for every
  $n\in\mathbb{N}$.  Let $\theta_{n\leftarrow
> n+1}:\mathbb{N}\to\mathbb{N},k\mapsto k+1$.  Now assume
  $(x_n)_{n\in\mathbb{N}}$ is contained in the inverse limit.  Then
  $x_{n+1}=x_n-1$ for every $n\in\mathbb{N}$ yielding a contradiction.

Reference
My question is: How is the inverse limit the empty set? I have computed that the inverse limit is the set of all constant sequences of $\mathbb N$.
Could someone explain this?  The author commented on it but I was unable to understand it.

The problem is that as the indices increase the numbers decrease by $1$. Eventually we will have to hit a negative number, but $\mathbb N$ does not contain any of those.
