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Computational scalability allows neural networks on embedded systems to provide desirable inference performance while satisfying severe power consumption and computational resource constraints. This paper presents a simple yet scalable inference method called ProgressiveNN, consisting of bitwise binary (BWB) quantization, accumulative bit-serial (ABS) inference, and batch normalization (BN) retraining. ProgressiveNN does not require any network structure modification and obtains the network parameters from a single training. BWB quantization decomposes and transforms each parameter into a bitwise format for ABS inference, which then utilizes the parameters in the most-significant-bit-first order, enabling progressive inference. The evaluation result shows that the proposed method provides computational scalability from 12.5% to 100% for ResNet18 on CIFAR-10/100 with a single set of network parameters. It also shows that BN retraining suppresses accuracy degradation of training performed with low computational cost and restores inference accuracy to 65% at 1-bit width inference. This paper also presents a method to dynamically adjust the bit-precision of the ProgressiveNN to achieve a better trade-off between computational resource use and accuracy for practical applications using sequential data with proximity resemblance. The evaluation result indicates that the accuracy increases by 1.3% with an average bit-length of 2 compared with only the 2-bit BWB network.