This paper presents a novel on-the-fly, black-box, property-checking through learning approach as a means for verifying requirements of recurrent neural networks (RNN) in the context of sequence classification. Our technique steps on a tool for learning probably approximately correct (PAC) deterministic finite automata (DFA). The sequence classifier inside the black-box consists of a Boolean combination of several components, including the RNN under analysis together with requirements to be checked, possibly modeled as RNN themselves. On one hand, if the output of the algorithm is an empty DFA, there is a proven upper bound (as a function of the algorithm parameters) on the probability of the language of the black-box to be nonempty. This implies the property probably holds on the RNN with probabilistic guarantees. On the other, if the DFA is nonempty, it is certain that the language of the black-box is nonempty. This entails the RNN does not satisfy the requirement for sure. In this case, the output automaton serves as an explicit and interpretable characterization of the error. Our approach does not rely on a specific property specification formalism and is capable of handling nonregular languages as well. Besides, it neither explicitly builds individual representations of any of the components of the black-box nor resorts to any external decision procedure for verification. This paper also improves previous theoretical results regarding the probabilistic guarantees of the underlying learning algorithm.