Question on inverse limits

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.

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.