{Are written} (gegraptai). Perfect passive indicative of
graph(935c), "have been written" by John. {That ye may believe}
(hina pisteu(8874)e). Purpose with hina and the present active
subjunctive of pisteu(935c), "that you may keep on believing." The
book has had precisely this effect of continuous and successive
confirmation of faith in Jesus Christ through the ages. {Jesus is
the Christ, the Son of God} (I(8873)ous estin ho Christos ho huios
tou theou). The man named Jesus is identical with the Messiah
(the Anointed One) as opposed to the Cerinthian separation of the
Jesus of history and the Christ (aeon) of theology. And the
Docetic notion of a phantom body for Jesus with no actual human
body is also false. Jesus is the Son of God with all that this
high term implies, the Logos of  Joh 1:1-18  (the Prologue).
"Very God of very God," Incarnate Revealer of God. But there is a
further purpose. {And that believing ye may have life in his
name} (kai hina pisteuontes z(9388)n ech(8874)e en t(9369) onomati autou).
Note present participle pisteuontes (continuing to believe) and
the present active subjunctive ech(8874)e (keep on having). "Life"
(z(9388)n) is eternal life so often mentioned in this Gospel, life
to be found only in the name (and power) of Jesus Christ the Son
of God. This verse constitutes a fitting close for this wonderful
book and John may at first have intended to stop here. But before
he published the work he added the Epilogue (Chapter XXI) which
is written in the same style and gives a beautiful picture of the
Risen Christ with a side-light on John and Peter (restored to
fellowship).