Oscar was a prince for a reason. After he had his doubts, he didn't go to confront Jasmine immediately. At first, he had given orders to find out what Jasmine was doing all the time. The maids all said the truth. To get on Oscar's good sides, they revealed everything including her ignorance to his daughters. One of the maids revealed that she saw Jasmine somewhere going each night which aroused the doubts in his mind.
Meanwhile, his guards searched everywhere where she could potentially go at night. Jasmine thought that she had covered it well but she was never above a prince's power. After all, Oscar was the owner of the castle and had access to everywhere.
