Based on the information provided, James is the killer. Here's how the detective resolved the case:
1. Jack and John's statements match: Both Jack and John confirm that they were at an official reception with their father that night and arrived home together. They also both state that their father entered the living room first, followed by Jack and John.
2. James' statement contradicts: James claims that he visited his friends that night and arrived home to find his father already dead. However, his statement contradicts Jack and John's statements, as they both state that their father entered the living room before James.
3. Inconsistency indicates guilt: The detective deduces that James is lying about his alibi and is the killer. His statement contradicts the sequence of events provided by Jack and John, suggesting that he is trying to cover up his involvement in the crime.
By comparing the statements of the three sons, the detective was able to identify James as the killer based on the inconsistency in his statement. The specific evidence or further actions taken by the detective to prove James' guilt are not provided in the given information.