The gun (used to kill Candy’s dog and also Lennie) is symbolic of the power used by those who decide that individuals are not of any value, whether it is the government or any kind of authority. Candy’s dog has become old and useless, therefore of no value. In the pragmatic world, it is only those who are useful and not a drain on society at large that are allowed to live and flourish. Lennie, who is “weak-minded” and unpredictable, as well as violent, also cannot be allowed to run free. George has been his protector, trying (and failing) to keep Lennie out of trouble. When Lennie kills Curley’s wife (who is also not worth being identified by a name), he also becomes a menace to society, even for George. As in the case with Candy’s dog, it is considered a good thing, even a kindness, to kill it. When George kills Lennie, it is also presented as the best thing for Lennie. He will always get in trouble. He will always be a danger to himself and others. He will always be a burden to someone. Best to just get rid of him.