The prediction of the Remaining Useful Life (RUL) is a critical step in Prognostics and Health Management (PHM) of systems under degradation. For efficient RUL predictions, most of the Artificial Intelligence (AI-based) methods perform direct mapping between raw sensor data input and RUL data as output targets for supervised learning. However, in the majority of the real-life cases, the available data are either incomplete or unlabeled, which calls for unsupervised methods. This paper proposes such an unsupervised RUL prediction method. Firstly, this method uses an autoencoder model to extract a Virtual Health Index (VHI) from sensors readings. Secondly, an LSTM-based (Long Short-Term Memory) encoder-decoder achieves VHI future predictions. Once the VHI prediction exceeds a pre-determined threshold, the RUL is recursively inferred. Such a method thus allows to obtain RUL predictions without using RUL-labeled data. This method is tested on C-MAPSS dataset. The results obtained are encouraging and offer new perspectives for real industrial applications