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The villain doesn't need to convince the protagonist as much as you need to convince the reader. If you can make it believeable to the reader that the protagonist changes sides, then it will be a...
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#3: Attribution notice added
Source: https://writers.stackexchange.com/a/40619 License name: CC BY-SA 3.0 License URL: https://creativecommons.org/licenses/by-sa/3.0/
#2: Initial revision
The villain doesn't need to convince the protagonist as much as # you need to convince the reader. If you can make it believeable _to the reader_ that the protagonist changes sides, then it will be a satisfying read. We often find out that our suspicions were mistaken in real life. For example, quite often a law enforcement agency finds out that their suspect was innocent. Why shouldn't that happen in a novel? But you might also show how the villain _manipulates_ the protagonist succesfully into believing an untruth. Protagonists don't always succeed, and this might be a story about a protagonist succumbing to some flaw. This too is something that happens in real life, where we often like to believe what we think _should_ be true, but in fact isn't. If you manage to narrate this process convincingly, the reader will gladly follow you to your unhappy ending.