"These things have I written unto you that believe on the name of The Son of God; that ye may know that ye have Eternal Life, and that ye may believe on the name of The Son of God." [1 John 5:13.]

It seems that John does a "double up" here, but yet, it is not a "double up" because, once we have believed on Jesus (as Paul says,  "we are not of them that draw back unto perdition, but of them that believe unto the saving of the soul" [Hebrews 10:39.]) there remains much room for the increase of this "Believing on Jesus".

When we first come to Christ, we believe on Him for salvation and the Gift of Eternal Life, which is what the apostle John is initially talking about but then adds this further encouragement to "Believe", and this second "Believe" is the "Believe" that ushers us into an ever growing experience of The Life of Christ, operating in us and through us.

Once we have believed in Jesus for salvation, this word of John is concerning believing on Christ that he is our,

This simply means, believing that, as Jesus said  "without me ye can do nothing" [John 15:5.] and as Solomon said  "All is vanity" [Ecclesiastes 2:17.] - That is not a part of the "All" that we are believing on Jesus for!

Paul sought to convey this wonderful truth to the Corinthian believers with the words,  "But of Him are ye in Christ Jesus, who of God is made unto us,                                                

                          And 

[1 Corinthians 1:30.]