So Peter and the other disciple started for the tomb. Both were running, but the other disciple outran Peter and reached the tomb first. He bent over and looked in at the strips of linen lying there but did not go in. Then Simon Peter, who was behind him, arrived and went into the tomb. He saw the strips of linen lying there, as well as the burial cloth that had been around Jesus’ head. The cloth was folded up by itself, separate from the linen. Finally the other disciple, who had reached the tomb first, also went inside. He saw and believed.
Why did John choose to mention three times that he beat Peter to the tomb?
There's significant discussion regarding the ending of John and how disjointed it seems (specifically, chapter 20). This has led some to hypothesize that John's gospel has been edited a few times. These could be insertions from the Johannine community revising an original document.– swasheck May 2 '12 at 21:36