The commercial interest in proximity services is increasing. Application examples include location-based information and advertisements, logistics, social networking, file sharing, etc. In this paper, we consider positioning of devices based on a time series of proximity reports from a mobile device to a network node. This corresponds to nonlinear measurements with respect to the device position in relation to the network nodes. Motion model will be needed together with the measurements to determine the position of the device. Therefore, sequential Monte Carlo methods, namely particle filtering and smoothing, are applicable for positioning. Positioning performance is evaluated in a typical office area with Bluetooth-low-energy beacons deployed for proximity detection and report, and is further compared to parametric Cramér-Rao lower bounds. Finally, the position accuracy is also evaluated with real experimental data.