Why was John's Gospel written? To give 'life' (John's Gospel for Skeptics 3)

One needless debate about 'John' is why he wrote his Gospel. 'Needless,' of course, because, like Luke, he informs his readers precisely why he wrote it. One may distrust him on other aspects of his book, but I see no reason to on this score. A book of this size and quality - being very carefully and thoughtfully composed - is a massive effort. The author clearly comes from the orthodox or proto-orthodox Christian tradition which itself also held that 'life' was the primary aim of Jesus' mission and Gospel. A more open debate is whether 'John' had a narrow or broader audience in mind (I think broader). The question I take up in this video is what exactly does John mean when he describes his book's purpose. As always, consult Ehrman among others to hear alternative perspectives.
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I have interviewed many non-believers for the podcast The Divide, where my good friend (an Atheist) and I (a Christian) co-interview people about the deeper issues that divide our culture. No surprise that many of the non-believing become rather severe critics of the Bible. Fair enough. Still, a number of these non-believing critics haven't, so to speak, watched the movie very 'critically.' This series aims to explain the Gospels directly to non-believing critics. To do so, the series will work on two levels:
1) The meaning of Gospel stories on their own terms
2) The grounds for why Christians take these stories seriously still today.
I am a classicist by training, who teaches classical languages and Great Books in university.