While monotonicity is a necessary and almost sufficient condition for Nash implementation and often a demanding one, almost any (non-monotonic, for instance) social choice rule can be implemented using undominated Nash or subgame perfect equilibrium.  By requiring solution concepts to have closed graph in the limit of complete information, Chung and Ely (2003) show that only monotonic social choice rules can be implemented in the closure of the undominated Nash equilibrium correspondence. In this paper, we show that only monotonic social choice rules can be implemented in the closure of the subgame perfect equilibrium/sequential equilibrium correspondence. Our robustness result help understand the limits of subgame pefect implementation, which is widely used in applications. We also argue that static mechanisms might outperform sequential mechanisms when one insists on robustness.