Is John the one?

Is John "the disciple who Jesus loved"?  This passage seems to indicate so.

John 21:20-25(NKJV)

Then Peter, turning around, saw the disciple whom Jesus loved following, who also had leaned on His breast at the supper, and said, “Lord, who is the one who betrays You?” 

Peter, seeing him, said to Jesus, “But Lord, what about this man?”

Jesus said to him, “If I will that he remain till I come, what is that to you? You follow Me.”

Then this saying went out among the brethren that this disciple would not die. Yet Jesus did not say to him that he would not die, but, “If I will that he remain till I come, what is that to you?”

This is the disciple who testifies of these things, and wrote these things; and we know that his testimony is true.

And there are also many other things that Jesus did, which if they were written one by one, I suppose that even the world itself could not contain the books that would be written. Amen.