= Solution
Use two expansions of the rational order. In the first interpret $c_n=n$, so the named sequence is unbounded above. In the second interpret $c_n=-1/n$, so the named sequence is strictly increasing but has upper bounds in the rational order. Both expansions are countably infinite <first-order models> of the specified <first-order theory>.
An <isomorphism> preserving each $c_n$ would preserve the existence of an element greater than all the named constants, since any such element in one <first-order structure> would map to such an element in the other. The first <first-order structure> has none and the second has, for instance, $0$. They cannot be <isomorphic>. Thus
$$
\boxed{\mathrm{DLO}_\omega\text{ is not }\aleph_0\text{-categorical}.}
$$
The property of lying above all $c_n$ is an infinite <partial type> rather than a single <first-order sentence>. This is why its realization can distinguish the two <first-order models> without contradicting the completeness established in (iv).
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