Learning the user-item relevance hidden in implicit feedback data plays an important role in modern recommender systems. Neural sequential recommendation models, which formulates learning the user-item relevance as a sequential classification problem to distinguish items in future behaviors from others based on the user's historical behaviors, have attracted a lot of interest in both industry and academic due to their substantial practical value. Though achieving many practical successes, we argue that the intrinsic {\bf incompleteness} and {\bf inaccuracy} of user behaviors in implicit feedback data is ignored and conduct preliminary experiments for supporting our claims. Motivated by the observation that model-free methods like behavioral retargeting (BR) and item-based collaborative filtering (ItemCF) hit different parts of the user-item relevance compared to neural sequential recommendation models, we propose a novel model-agnostic training approach called WSLRec, which adopts a three-stage framework: pre-training, top-$k$ mining, and fine-tuning. WSLRec resolves the incompleteness problem by pre-training models on extra weak supervisions from model-free methods like BR and ItemCF, while resolves the inaccuracy problem by leveraging the top-$k$ mining to screen out reliable user-item relevance from weak supervisions for fine-tuning. Experiments on two benchmark datasets and online A/B tests verify the rationality of our claims and demonstrate the effectiveness of WSLRec.