But these are written, that ye might believe that Jesus is the Christ, the Son of God; and that believing ye might have life through his name.
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John 20:29-31
Tyndale Open Study Notes (CC BY-SA 4.0).
20:31 that you may continue to believe: The present tense (found in some manuscripts) indicates that John wrote to encourage believers. Other manuscripts read that you may [come to] believe, suggesting that John wrote to stimulate new faith (cp. 1:7).
Other Bible passages linked to this one, most cited first.
To him give all the prophets witness, that through his name whosoever believeth in him shall receive remission of sins.
That whosoever believeth in him should not perish, but have eternal life. For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.
Whosoever believeth that Jesus is the Christ is born of God: and every one that loveth him that begat loveth him also that is begotten of him.
Whosoever shall confess that Jesus is the Son of God, God dwelleth in him, and he in God.
And we know that the Son of God is come, and hath given us an understanding, that we may know him that is true, and we are in him that is true, even in his Son Jesus Christ. This is the true God, and eternal life.
He that believeth on the Son hath everlasting life: and he that believeth not the Son shall not see life; but the wrath of God abideth on him.
Authorship: the book itself or Christian tradition. Approximate dates, as in Carson and Moo, An Introduction to the New Testament, and Longman and Dillard, An Introduction to the Old Testament.