The degradation of aeroengines can be divided into two stages: the healthy stage and the unhealthy stage. Remain useful life (RUL) prediction should be triggered from the start time of the unhealthy stage to ensure safe operation. Nevertheless, many existing RUL prediction methods simply assign a fixed DSP to any aeroengine, limiting further improvement as the DSP is uncertain and varies with individual differences of aeroengines. To address this issue, a novel two-stage deep residual long-short term memory (Dual-DRLSTM) is developed, which integrates DSP detection and RUL prediction into one framework, and associates them through degradation health index (HI). First, DRLSTM is employed as the backbone to extract representative degradation features from multi-dimensional time-series monitoring data. Second, the Dual-DRLSTM relaxes the strong assumption of the fixed degradation start point (DSP) and performs DSP detection for each aeroengine. Then, the Dual-DRLSTM predicts the RUL of the aeroengine beyond the DSP in the unhealthy stage. Finally, the outstanding performance of Dual-DRLSTM is validated through a series of experiments on a public C-MAPSS dataset.