Recent empirical works show that large deep neural networks are often highly redundant and one can find much smaller subnetworks without a significant drop of accuracy. However, most existing methods of network pruning are empirical and heuristic, leaving it open whether good subnetworks provably exist, how to find them efficiently, and if network pruning can be provably better than direct training using gradient descent. We answer these problems positively by proposing a simple greedy selection approach for finding good subnetworks, which starts from an empty network and greedily adds important neurons from the large network. This differs from the existing methods based on backward elimination, which remove redundant neurons from the large network. Theoretically, applying the greedy selection strategy on sufficiently large {pre-trained} networks guarantees to find small subnetworks with lower loss than networks directly trained with gradient descent. Our results also apply to pruning randomly weighted networks. Practically, we improve prior arts of network pruning on learning compact neural architectures on ImageNet, including ResNet, MobilenetV2/V3, and ProxylessNet. Our theory and empirical results on MobileNet suggest that we should fine-tune the pruned subnetworks to leverage the information from the large model, instead of re-training from new random initialization as suggested in \citet{liu2018rethinking}.