I write these things TO YOU WHO BELIEVE in the name of the Son of God that you may 'know' that you have eternal life.... 1 John 5:13
I think it's very interesting that the first verse above leans toward those who might not yet believe, (the use of 'may' could indicate some of the hearers hadn't yet believed though listening and wanting to know the truth). ..... it would also appear to encourage those who do believe.
The other, John 5:13, is definately for those who do believe so that they may "KNOW" they have eternal life.
Either way the scriptures attest that the scriptures can and do lead people to the truth they desire, and for that come to realize they need a Savior... that once they accept His forgiveness, freely given, the scriptures will confirm to them that they are indeed saved and their security in Christ sealed.
Jesus often explained the meaning of the Scripture from which they had strayed from...stating... “have you not read,” ....“it is written,” .....“For I say to you,” etc. ... and here Paul states...“Holding fast the faithful word as he has been taught, that he may be able, by sound doctrine, both to exhort and convict those who contradict” (Titus 1:9)
If someone cannot see a teacher is off on the fundamentals when they are shown the problem they are unable to receive from Scripture, they have not been trained by the word. ......'Holding fast the word' is someone who is Bible based,.. sound in his understanding of what the Scripture says and does not say... He understands what God has said as God’s word is living and active today just as it was when it was spoken or written.... http://www.letusreason.org/Doct56.htm
Especially John says that he did not write down everything Jesus did but selected what was written so that people could believe.
He says, BUT THESE THINGS were written so that you may believe.
IOW, John, under the inspiration of the Holy Spirit, wrote what God the Holy Spirit knew would be enough testimony about Jesus to lead someone to faith in Jesus.