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John 20:31
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. (John 20:31)
These are written.
John here sums up the purpose of his writing and the plan in his selection of material. It was not his aim to give a full history or even a detailed biography of Jesus. He chose those “signs” that formed the underlying basis of his theme and purpose in writing.
Jesus is the Christ.
 Jesus was the name assumed by Christ in His humanity (see on Matt. 1:21). It was His personal name, the name by which He was known to His contemporaries. To many the name identified merely the carpenter’s son. John’s purpose was to show that the Jesus men knew was indeed the Messiah. On “Christ” meaning “Messiah” see on Matt. 1:1.
Son of God.
 See on Luke 1:35.
Life.
 Gr. zōē (see on chs. 1:4; 8:51; 10:10). See ch. 6:47; see on ch. 3:16.