We see several examples of how Lennie tries to be like George, from the various points where he repeats what George says right after he says it (cussing at the bus driver, for example), to how he tries to work hard, follow George's rules and generally please him. It's very clear that not only does Lennie look up to George, he looks at him as a father or brother figure.
Lennie is, in fact, a child, at least mentally and emotionally. He is not capable of maturing. He has issues with his memory. George has to tell him over and over about the rabbits, not only because Lennie likes the story, but because he forgets the details all of the time. Steinbeck makes it very clear, from the beginning, how dependent Lennie is on George, and we get the uneasy feeling that even with George looking out for him, it's not going to be enough.