Given a set with a partial order $\leq$, can we say that the following is an infinite descending chain?
$a\geq\cdots a_{-2}\geq a_{-1}\geq a_{0}\geq a_1\geq a_{2}\cdots$
I am confused because I have normally seen infinite descending chains indexed by the set of natural numbers, $\mathbb{N}$ not by the set of integers, $\mathbb{Z}$ as above.
If $I$ is an index set, with linear order $<_I$ and we have $a_i \in (X,\le_X)$ for all $i \in I$ then we have a decreasing chain $(a_i)_{i \in I}$ iff
$$\forall i,j \in I: (i \le_I j) \implies a_i \ge_X a_j$$
so that larger-indexed elements are smaller. So under that definition, yes you have a decreasing chain, though the first element has no index, BTW and if you index by $\Bbb Z$ we will have no first-indexed element of cours, so leave it out.