The two were running together, but the other disciple outran Peter and reached the tomb first.
There was no one thing of which the apostles were more concerned to produce substantial proof than the resurrection of their Master, 1. Because it was that which he himself appealed to as the last and most cogent proof of his being the Messiah. Those that would not believe other signs were referred to this sign of the prophet Jonas.
Commenting on John 20:1-10
So they ran both together,.... At first setting out, and for a while; not content to walk, they ran, being eagerly desirous to know the truth of things: and that other disciple did outrun Peter, and came first to the sepulchre; John was a younger man than Peter, and so more nimble, and swift of foot, and got to the sepulchre before him; and besides...
Outrun Peter - Not because he had a greater desire to see into the truth of these things; but because he was younger, and lighter of foot.