Lit., I wrote. John speaks as looking back over his Epistle and recalling the aim with which he wrote. See on 1Jo 2:13.
May know (εἰδῆτε)
Not perceive (γινώσκειν), but know with settled and absolute knowledge. See on Joh 2:24.
Ye have eternal life (ζωὴν ἔχετε αἰώνιον)
The Greek order is peculiar, “ye may know that life ye have eternal.” The adjective eternal is added as an after-thought. So Westcott: “that ye have life - yes, eternal life.”
Unto you that believe
In the A.V., these words follow have I written. The Rev. follows the Greek order. The words, like eternal, above, are added as an after-thought, defining the character of the persons addressed.