What convinced Peter and John that Jesus was risen from the dead?
Peter and John were convinced by seeing the empty tomb and the evidence there, which showed Jesus must have been supernaturally raised from the dead. At that point they believed and understood the Scriptures for the first time, as recorded in John's Gospel chapter 20.
Les explains that on the resurrection morning, Mary Magdalene found the tomb empty and quickly ran to tell Peter and John, who evidently had spent the night not too far away. Both men ran to the sepulcher. When they saw the tomb was empty, they saw the evidence that He must have supernaturally been raised from the dead. Then, for the first time, they believed and understood the Scriptures. So the empty tomb and the evidence it presented convinced Peter and John that Jesus was risen.