Decreasing sequence of non-negative integers

Well, this seems kind of obvious, but let's do it anyway. We know three things:

1: The sequence is strictly decreasing, i.e, $a_{n+1}<a_n.$

2: $a_n in mathbb{Z} ~~forall n.$

3: $a_ngeq 0 ~forall n$.

Condition (1) implies that all of the terms of $a_n$ are distinct, i.e, $nexists m,ninmathbb{N}:a_m=a_n$. Now using condition (2), let $a_0=N$, some positive integer. We know, using condition (1), that all subsequent terms are $<N$, and using condition (3), are all $geq0.$ Now suppose $a_n$ had infinitely many terms. That would imply that there are infinitely many integers in the range $[0,N]$, which is clearly false. Thus $a_n$ can only have finitely many terms.

Can we ever have this kind of sequence ? As it is strictly monotonically decreasing so we have $a_1 geq a_n geq 0$ for all $ ngeq 2 $ but there are only finitely many non negative integers satisfying this .so the sequence is eventually constant which contradicts the fact it is strictly monotonically decreasing.