Well, the evidence is overwhelming from Scripture and Tradition. You can start with the fact, that while the other three gospels tell us John is an apostle, and one of the three most important apostles (e.g. Luke 9:28), the Gospel of John never mentions him by name. And another very good reason to believe he is John is this:
John 21 (NIV)
20Peter turned and saw that the disciple whom Jesus loved was following them. (This was the one who had leaned back against Jesus at the supper and had said, “Lord, who is going to betray you?”) 21When Peter saw him, he asked, “Lord, what about him?”
22Jesus answered, “If I want him to remain alive until I return, what is that to you? You must follow me.” 23Because of this, the rumor spread among the brothers that this disciple would not die. But Jesus did not say that he would not die; he only said, “If I want him to remain alive until I return, what is that to you?”
24This is the disciple who testifies to these things and who wrote them down. We know that his testimony is true.
And then we see:
Revelation 1 (NIV)
9I, John, your brother and companion in the suffering and kingdom and patient endurance that are ours in Jesus, was on the island of Patmos because of the word of God and the testimony of Jesus. 10On the Lord’s Day I was in the Spirit, and I heard behind me a loud voice like a trumpet, 11which said: “Write on a scroll what you see and send it to the seven churches: to Ephesus, Smyrna, Pergamum, Thyatira, Sardis, Philadelphia and Laodicea.”
12I turned around to see the voice that was speaking to me. And when I turned I saw seven golden lampstands, 13and among the lampstands was someone “like a son of man,”**dressed in a robe reaching down to his feet and with a golden sash around his chest. 14His head and hair were white like wool, as white as snow, and his eyes were like blazing fire. 15His feet were like bronze glowing in a furnace, and his voice was like the sound of rushing waters. 16In his right hand he held seven stars, and out of his mouth came a sharp double-edged sword. His face was like the sun shining in all its brilliance.
17When I saw him, I fell at his feet as though dead. Then he placed his right hand on me and said: “Do not be afraid. I am the First and the Last. 18I am the Living One; I was dead, and behold I am alive for ever and ever!”
So you see, Jesus returned to see St. John, the author of both his Gospel and Revelation, before he died - as Jesus foretold. Your professor, scoffer that he is, will probably say that these two books have different authors because the style of writing is different. Don’t worry about that, just remember that God is the ultimate Author of Sacred Scripture. So hold fast to Tradition, and also keep this in mind:
“Where is the wise man? Where is the scholar? Where is the philosopher of this age? Has not God made foolish the wisdom of the world?” (1 Corinthians 1:20)