The Gospel of John doesn't begin with an 'argument'. It begins with an explanation, drawn from several scriptures, namely Genesis 1 and Proverbs 8, among others. John 1:1, is an expounding upon Genesis 1:1,3,6-7, etc; Proverbs 8:22-36, and he reissues that explanation in 1 John 1:1-3, as in John 1:1-3.
The "son of God" is not being 'proven' by John's statements, but is explaining such verses, like as Zechariah 13:7, and so on. The Son of the Father is already present in Genesis 1, and John chapter 1, explains it. John calls Jesus the "Word", and this also (besides Genesis 1:1,&c "aleph tav") hearkens back to Genesis 3, notice:
Gen 3:8 And they heard the voice of the LORD God walking in the garden in the cool of the day: and Adam and his wife hid themselves from the presence of the LORD God amongst the trees of the garden.
"the voice of" is the "Word of" the Father, or in other words the person who came from the Father that represents or presents (as ambassador or highest messenger) the Father's will to others.
John 1 is not attempting to prove the Son of God's existence, but accepts it as underlying fact, based upon the previous scriptures that were accepted as cannon and true by the Hebrews/Israelites/Jews themselves.