Did you see how sickly Lee looked towards the end of the game? It looked like his eyes were turning yellow and his skin colour looked almost dead. There's no hope for him, it'd be unrealistic to think there was. He was bitten. This is the Walking Dead, people who are bitten die.
(Sure the chopping of the arm may have bought him some time if he would have did it right away, but it wasn't done immediately).
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