 Stationary partition at a successor cardinal

Every stationary subset of a successor cardinal ${\lambda }^{+}$ can be partitioned into ${\lambda }^{+}$ disjoint stationary subsets. For each $\beta \in [\lambda ,{\lambda }^{+})$ choose a bijection $e_{\beta }:\lambda \to \beta$. For each $\xi <{\lambda }^{+}$, the function $\beta \mapsto e_{\beta }^{-1}(\xi )$ is regressive on the stationary tail with $\beta >\xi$. The Fodor lemma gives a stationary fiber with one value $i_{\xi }<\lambda$. Some value occurs for ${\lambda }^{+}$ many indices, whose fibers are pairwise disjoint. Assign any leftover part of $S$ to one fiber.