On the first day of the week, Mary of Magdala came to the tomb early in the morning, while it was still dark, and saw the stone removed from the tomb. So she ran and went to Simon Peter and to the other disciple whom Jesus loved, and told them, "They have taken the Lord from the tomb, and we don't know where they put him." So Peter and the other disciple went out and came to the tomb. They both ran, but the other disciple ran faster than Peter and arrived at the tomb first; he bent down and saw the burial cloths there, but did not go in. When Simon Peter arrived after him, he went into the tomb and saw the burial cloths there, and the cloth that had covered his head, not with the burial cloths but rolled up in a separate place. Then the other disciple also went in, the one who had arrived at the tomb first, and he saw and believed. (Jn 20:1-8)
John the Evangelist calls himself as ‘the other disciple whom Jesus loved’. It is an interesting description of himself and this is how he wants to be remembered. This shows the relationship which he has with Jesus or the relationship which he wants to have with Jesus. Each one should have his own definition or description of his relationship with Jesus and that is what he should reveal to the others and not his personal identity.
Peter and John ran. John ran faster than Peter, may be because he was young or he was good at running. In the garden of Gethsemane the disciples were running away from Jesus (all, everyone). Here they are running towards Jesus. John, though he ran faster he did not make use of his talents to gain precedence. He waited for the other.
At the end he also entered in and he saw the empty tomb and he believed. His belief was based on the empty tomb. Belief in the resurrection comes from the empty tomb. Emptiness is the sign of hope. Nothingness fills their mind with presence, a powerful total presence; presence of Jesus.