How can I prove that George is innocent of killing Lennie in the book Of Mice and Men?
First consider different definitions/interpretations of the concept of innocence. It's hard to prove that George is legally innocent of killing Lennie because, quite simply, he does kill Lennie and admits to it. And George is not innocent in the sense that he is naive and didn't know what he was doing.
But we can present an argument that shows George did the right thing, considering the circumstances. Therefore, he is free of wrongdoing. In other words, George kills Lennie out of an act of mercy. There is no evil intent or motive and he kills him to prevent further violence and/or death to Lennie himself and, potentially, to others that Lennie might accidentally hurt in the future (as he'd hurt Curley's wife).
When Candy shows Curley's wife's dead body to George, George says "I should of knew" and "I guess maybe way back in my head I did." Even though George takes some responsibility for this tragedy with this statement, it is too much to suggest that he is guilty. In fact, it is remarkable that he's kept Lennie from committing a tragedy such as this for so long.
George hopes that they will put Lennie in prison. At least, he will be fed and looked after. But then Curley swears he's going to kill Lennie. Curley then says to shoot for Lennie's guts. This would be extremely painful. In the end, George kills Lennie to save him from a painful, fearful end with Curley. It is a mercy killing.
In fact, in the end, George tells Lennie about their bond and about the rabbits just before shooting him. Lennie dies in the most peaceful way, considering the circumstances, with the dream of the farm in his mind. If Curley is determined to kill Lennie in a painful way, George saves Lennie from such a horrible end. Even Slim, the most fair-minded worker on the ranch, agrees that George did what was necessary. He says "You hadda, George. I swear you hadda. Come on with me." You could even argue that this was euthanasia, which is defined as ending a life in order to relieve or avoid pain and suffering. In killing Lennie in this way, he avoids the painful, terrifying (for Lennie) death that would await Lennie when Curley got to him.