A.T. Robertson Commentary 1 John 5:13

"These things have I written unto you, that ye may know that ye have eternal life, [even] unto you that believe on the name of the Son of God." — 1 John 5:13 (ASV)

I have written (εγραψα). Not epistolary aorist, but refers to verses 1-12 of this Epistle as in 2:26 to the preceding verses.

That ye may know (ινα ειδητε). Purpose clause with ινα and the second perfect active subjunctive of οιδα, to know with settled intuitive knowledge. He wishes them to have eternal life in Christ (John 20:31) and to know that they have it, but not with flippant superficiality (2:3ff.).

Unto you that believe on (τοις πιστευουσιν εις). Dative of the articular present active participle of πιστευω and εις as in verse 10. For this use of ονομα (name) with πιστευω see 3:23; John 2:23.