The delay constrained relay node placement (DCRNP) problem minimizes the quantity of deployed relay nodes that are employed to build at least one path between a sink and each sensor node and guarantees that the delay constraints for the built paths are fulfilled. It has been proven that the DCRNP problem is NP-hard. This paper proposes a convergence-pruning-based relay node placement (CPRNP) algorithm to approximately solve the DCRNP problem. The CPRNP algorithm consists of two stages. In the first stage, CPRNP identifies the convergences of all the paths meeting the delay constraint and forms a shortest-path tree that is rooted at the sink and connects all the sensors. In the second stage, the CPRNP gradually reduces the number of deployed relay nodes by deleting or substituting the nodes on the shortest-path tree. The simulation results confirm that CPRNP can significantly save deployed relay nodes compared to existing algorithms.