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Unsupervised domain adaptation reduces the reliance on data annotation in deep learning by adapting knowledge from a source to a target domain. For privacy and efficiency concerns, source-free domain adaptation extends unsupervised domain adaptation by adapting a pre-trained source model to an unlabeled target domain without accessing the source data. However, most existing source-free domain adaptation methods to date focus on the transductive setting, where the target training set is also the testing set. In this paper, we address source-free domain adaptation in the more realistic inductive setting, where the target training and testing sets are mutually exclusive. We propose a new semi-supervised fine-tuning method named Dual Moving Average Pseudo-Labeling (DMAPL) for source-free inductive domain adaptation. We first split the unlabeled training set in the target domain into a pseudo-labeled confident subset and an unlabeled less-confident subset according to the prediction confidence scores from the pre-trained source model. Then we propose a soft-label moving-average updating strategy for the unlabeled subset based on a moving-average prototypical classifier, which gradually adapts the source model towards the target domain. Experiments show that our proposed method achieves state-of-the-art performance and outperforms previous methods by large margins.