Often times, a motivating factor in striving for some goal or some kind of success is proving someone else wrong. People in high school who thought you were a loser. A teacher or professor who said you were a terrible writer. A parent. A sibling. Whoever.
Most of the time, though, I don't think these people really care if we ever make our goal. We haven't really proven anything--if a professor thought you were a terrible writer, and you manage to get published, that professor's mind probably hasn't changed. Now you're just a terrible published writer, in that professor's eyes--if he or she even cares. If those people from high school haven't grown up (and some of them never do) you're still a loser, no matter what you do. (I think the exception to this is parents. But I digress.)
I don't think we're proving them wrong. I don't think we're even striving to prove them wrong. I think they become the face of our own doubt. Someone says you're a bad writer? Now you have a picture to hang on the dartboard--but that feeling of doubt you want to throw darts at has probably always been with you in some capacity. These people certainly don't help matters. They make the doubt worse. Or they plant a different kind of doubt, now to grow wild among all the other kinds already living within us.
When we say we want to prove them wrong, I think it's really doubt we want to prove wrong. Our own doubt. My doubt has a lot of faces, collected over the years from people who made me feel small. But the person I most want to prove wrong is myself.