John begins the Gospel by describing the Word who was with God, came to earth, and returned to His Father:
1In the beginning was the Word, and the Word was with God, and the Word was God. 2He was in the beginning with God. 18No one has ever seen God; the only God, who is at the Father's side, he has made him known. (John 1)
1ἐν ἀρχῇ ἦν ὁ λόγος καὶ ὁ λόγος ἦν πρὸς τὸν θεόν καὶ θεὸς ἦν ὁ λόγος 2οὗτος ἦν ἐν ἀρχῇ πρὸς τὸν θεόν 18θεὸν οὐδεὶς ἑώρακεν πώποτε μονογενὴς θεὸς ὁ ὢν εἰς τὸν κόλπον τοῦ πατρὸς ἐκεῖνος ἐξηγήσατο
The Gospel states the Word is the Son of God, Jesus Christ who was sent. It then narrates how the Son manifested God and brought to mankind the opportunity to obtain eternal life. Based on the Prologue, the reader understands the narrative ends when the Son is reunited with the Father. John's first letter ends with the same summary:
And we know that the Son of God has come and has given us understanding, so that we may know him who is true; and we are in him who is true, in his Son Jesus Christ. He is the true God and eternal life. (1 John 5:20)
οἴδαμεν δὲ ὅτι ὁ υἱὸς τοῦ θεοῦ ἥκει καὶ δέδωκεν ἡμῖν διάνοιαν ἵνα γινώσκωμεν τὸν ἀληθινόν καὶ ἐσμὲν ἐν τῷ ἀληθινῷ ἐν τῷ υἱῷ αὐτοῦ Ἰησοῦ Χριστῷ οὗτός ἐστιν ὁ ἀληθινὸς θεὸς καὶ ζωὴ αἰώνιος
The Son of God who was sent is now historically the Son of God who came (i.e the Gospel narrative) and has given us understanding (1:18) so we may know His who is true (and is reunited with the Father).
What strikes me is how the initial relationship "in the beginning" before coming refers to the Word as οὗτος and that is how the letter describes the present-day working relationship of the Son of God and God to bring eternal life. In other words, the only difference between John 1:1-2 and 1 John 5:20 is the physical coming which is now a past event. The use of οὗτος in John 1:2 and then again to end the letter seems to be purposeful.
Does οὗτός at the end of the letter form an inclusio to the beginning of the Gospel?