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 constant sequences $(c_n)_{n\in\Bbb{N}}$ of $\Bbb{N}$ are not in the inverse limit because they do not satisfy
$$\theta_{n\leftarrow n+1}(c_{n+1})=c_n,$$
for any $n$ at all. To see that the inverse limit is empty, suppose toward a contradiction that $(x_n)_{n\in\Bbb{N}}$ is in the inverse limit. Then it satisfies $x_n=\theta_{n\leftarrow n+1}(x_{n+1})$ for all $n\in\Bbb{N}$, and by induction
$$x_n=\theta_{n\leftarrow n+m}(x_{n+m})=x_{n+m}+m,$$
holds for all $,\in\Bbb{N}$. Taking $n=1$ and $m=x_1$ we get
$$x_1=\theta_{1\leftarrow\ 1+x_1}(x_{1+x_1})=x_{1+x_1}+1+x_1,$$
a contradiction.
Colloquially speaking, the fact that $x_{n+1}=x_n-1$ means that the sequence is strictly decreasing. An infinite sequence in $\Bbb{N}$ cannot be strictly decreasing, so there is no sequence in the inverse limit.
