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Image super-resolution (SR) technique can improve the spatial resolution of images without upgrading the imaging system. As a result, SR promotes the development of high resolution (HR) remote sensing image applications. Many remote sensing image SR algorithms based on deep learning have been proposed recently, which can effectively improve the spatial resolution under the constraints of HR images. However, images acquired by remote sensing imaging devices typically have lower resolution. Hence, an insufficient number of HR remote sensing images are available for training deep neural networks. In view of this problem, we propose an unsupervised SR method that does not require HR remote sensing images. The proposed method introduces a generative adversarial network (GAN) that obtains SR images through the generator; then, the SR images are downsampled to train the discriminator with low resolution (LR) images. Our method outperformed several methods in terms of the quality of the obtained SR images as measured by 6 evaluation metrics, which proves the satisfactory performance of the proposed unsupervised method for improving the spatial resolution of remote sensing images.