We introduce a general self-supervised approach to predict the future outputs of a short-range sensor (such as a proximity sensor) given the current outputs of a long-range sensor (such as a camera). We assume that the former is directly related to some piece of information to be perceived (such as the presence of an obstacle in a given position), whereas the latter is information rich but hard to interpret directly. We instantiate and implement the approach on a small mobile robot to detect obstacles at various distances using the video stream of the robot's forward-pointing camera, by training a convolutional neural network on automatically-acquired datasets. We quantitatively evaluate the quality of the predictions on unseen scenarios, qualitatively evaluate robustness to different operating conditions, and demonstrate usage as the sole input of an obstacle-avoidance controller. We additionally instantiate the approach on a different simulated scenario with complementary characteristics, to exemplify the generality of our contribution.