If you feel the need to prove yourself to them, then you are probably not the one who is better.
I see no need to prove that you are better than anyone. If you are good at something, better than others, that is a plus for you, and everyone will be able to see that clearly even without you showing it.
The constant need to prove yourself makes you an asshole, and even if you’re the best at something, you’ll still be seen as an asshole. That kind of proof can only do more harm than good.