Pruning techniques for neural networks with a recurrent architecture, such as the recurrent neural network (RNN), are strongly desired for their application to edge-computing devices. However, the recurrent architecture is generally not robust to pruning because even small pruning causes accumulation error and the total error increases significantly over time. In this paper, we propose an appropriate pruning algorithm for RNNs inspired by "spectral pruning", and provide the generalization error bounds for compressed RNNs. We also provide numerical experiments to demonstrate our theoretical results and show the effectiveness of our pruning method compared with existing methods.