I have already adduced several proofs of Jesus' divinity, but I will add to them. One of God's many attributes is that he is all-encompassing. Scripture puts it this way:
Isaiah 41:4 Who hath wrought and done it, calling the generations from the beginning? I the LORD, the first, and with the last; I am he.
Isaiah 44:6 Thus saith the LORD the King of Israel, and his redeemer the LORD of hosts; I am the first, and I am the last; and beside me there is no God.
Isaiah 48:12 Hearken unto me, O Jacob and Israel, my called; I am he; I am the first, I also am the last.
Jesus is also identified by this same description. We find this in several places, both directly and also in a parallel Greek expression.
Revelation 1:8 I am Alpha and Omega, the beginning and the ending, saith the Lord, which is, and which was, and which is to come, the Almighty.
Revelation 1:11 Saying, I am Alpha and Omega, the first and the last: and, What thou seest, write in a book, and send it unto the seven churches which are in Asia; unto Ephesus, and unto Smyrna, and unto Pergamos, and unto Thyatira, and unto Sardis, and unto Philadelphia, and unto Laodicea.
Revelation 1:17 And when I saw him, I fell at his feet as dead. And he laid his right hand upon me, saying unto me, Fear not; I am the first and the last:
Revelation 2:8 And unto the angel of the church in Smyrna write; These things saith the first and the last, which was dead, and is alive;
Revelation 21:6 And he said unto me, It is done. I am Alpha and Omega, the beginning and the end. I will give unto him that is athirst of the fountain of the water of life freely.
Revelation 22:13 I am Alpha and Omega, the beginning and the end, the first and the last.
I realize that modern versions shorten the expression in Revelation 1:8 and omit the relevant expression in Revelation 1:11, but they have the others. My point here is not to criticize the modern versions but to simply note that both the KJV and the modern versions contain this same proof.
The connection to Jesus is fairly clear. The passage leading up Revelation 1:17 states:
And in the midst of the seven candlesticks one like unto the Son of man, clothed with a garment down to the foot, and girt about the paps with a golden girdle. His head and his hairs were white like wool, as white as snow; and his eyes were as a flame of fire; and his feet like unto fine brass, as if they burned in a furnace; and his voice as the sound of many waters. And he had in his right hand seven stars: and out of his mouth went a sharp twoedged sword: and his countenance was as the sun shineth in his strength.
He is the one addressing John and instructing him to write to the seven churches (which includes the item at Revelation 2:8).
An additional clue is that the book is the Revelation (note that it is singular) of Jesus Christ.
The Revelation of Jesus Christ, which God gave unto him, to shew unto his servants things which must shortly come to pass; and he sent and signified it by his angel unto his servant John: who bare record of the word of God, and of the testimony of Jesus Christ, and of all things that he saw.
John to the seven churches which are in Asia: Grace be unto you, and peace, from him which is, and which was, and which is to come; and from the seven Spirits which are before his throne; and from Jesus Christ, who is the faithful witness, and the first begotten of the dead, and the prince of the kings of the earth. Unto him that loved us, and washed us from our sins in his own blood, and hath made us kings and priests unto God and his Father; to him be glory and dominion for ever and ever. Amen. Behold, he cometh with clouds; and every eye shall see him, and they also which pierced him: and all kindreds of the earth shall wail because of him. Even so, Amen.
Similarly, in Revelation the speaker is described in this context:
Rev 21:5 And he that sat upon the throne said, Behold, I make all things new. And he said unto me, Write: for these words are true and faithful.
Rev 21:6 And he said unto me, It is done. I am Alpha and Omega, the beginning and the end. I will give unto him that is athirst of the fountain of the water of life freely.
The reference to the fact that Jesus is the enthroned one would be enough, perhaps, but notice that there is also a reference back to the water of life, which Jesus described this way:
John 4:14 But whosoever drinketh of the water that I shall give him shall never thirst; but the water that I shall give him shall be in him a well of water springing up into everlasting life.
We see this water of life again in the final reference in Revelation:
And, behold, I come quickly; and my reward is with me, to give every man according as his work shall be. I am Alpha and Omega, the beginning and the end, the first and the last. Blessed are they that do his commandments, that they may have right to the tree of life, and may enter in through the gates into the city. For without are dogs, and sorcerers, and whoremongers, and murderers, and idolaters, and whosoever loveth and maketh a lie. I Jesus have sent mine angel to testify unto you these things in the churches. I am the root and the offspring of David, and the bright and morning star. And the Spirit and the bride say, Come. And let him that heareth say, Come. And let him that is athirst come. And whosoever will, let him take the water of life freely.
Praise be to the Alpha and Omega, the beginning and the end, the first and the last, even the Lord Jesus Christ,