The complete-metric Baire principle implies Dependent Choice over ZF

Since $A\mathrm{\ne }\mathrm{\varnothing }$, $Y=A^{\omega }$ with its specified metric is nonempty complete, and $(U_i)$ is open dense. Apply CM-Baire to this space and family: $D={\bigcap }_i{U}_i$ is dense in $Y$. If $D$ were empty, every ball about a point of the nonempty space $Y$ would miss it, contradicting density. Thus fix a single $f\in D$.

For each $i\in \omega$, Separation gives $W_i=\{j\in \omega :f(i)Rf(j)\}$. Since $f\in U_i$, this set is nonempty and has a unique least element $q(i)$. The graph of $q$ is the subset of $\omega \times \omega$ where $j\in W_i$ and no smaller natural belongs to $W_i$; hence Separation makes $q:\omega \to \omega$ a set function. In particular $f(i)Rf(q(i))$ for every $i$. This defines successors uniquely from the one fixed $f$.

Apply recursion with carrier $\omega$, initial element $0$ and the self-map $q$. It yields $k:\omega \to \omega$ with $k(0)=0$ and $k(n+1)=q(k(n))$. The composite $a(n)=f(k(n))$ has graph obtained by Separation in $\omega \times A$. For every $n$, the preceding relation at $i=k(n)$ says $a(n)=f(k(n))Rf(q(k(n)))=f(k(n+1))=a(n+1)$. Thus $a$ is an $R$-chain.

The construction works for every nonempty $A$ and every serial $R$, so gives the global starting-point-free DC principle. Its ZF equivalence with the prescribed-start principle gives the latter as well. The minimum $q(i)$ can be smaller or larger than $i$, so no increasing-index or distinct-value assumption entered the argument; singleton carriers and self-loops are allowed.