We prove a multi-channel telescoping sum boosting theory for the ResNet architectures which simultaneously creates a new technique for boosting over features (in contrast to labels) and provides a new algorithm for ResNet-style architectures. Our proposed training algorithm, BoostRes-Net, is particularly suitable in non-differentiable architectures. Our method only requires the relatively inexpensive sequential training of T "shallow ResNets". We prove that the training error decays exponentially with the depth T if the weak module classifiers that we train perform slightly better than some weak baseline. In other words, we propose a weak learning condition and prove a boosting theory for ResNet under the weak learning condition. A generalization error bound based on margin theory is proved and suggests that ResNet could be resistant to overfitting using a network with l1 norm bounded weights.