Which $c_n$ are you referring to? My construction refers to all of them (or at least infinitely many of them).In this case, $c_n$ would have to equal $x$ (where $n$ is sufficiently large as you state). If it didn't, the sets $A$ and $B$ would not form a Dedekind cut for $x$. Rather, they would form a Dedekind cut for $c_n$.
The construction is similar to the definition of the limit. For each $a < x$ we can find an $N$ such that $a < c_n$ for all $n > N$ precisely because for each $\epsilon > 0$ there exists an $N$ such that $(x - c_n) < \epsilon$ for all $n > N$. (Note that $(x-c_n)$ is non-negative because the sequence $(c_n)$ is monotonically increasing with limit $x$). Thus, for every $a < x$ we can pick $\epsilon < (x-a)$ and thus find our value of $N$. "All sufficiently large $n$" then refers to "all $n > N$".