The very first thing to understand about “proving yourself”, is that YOU need to be the first person you convince. You need to develop your abilities hand-in-hand with your confidence to the point where you feel pride in what you can do. Once you reach that point, you can then work on impressing other people. A word of warning, though: the point of artistic pursuit is not, and never has been, to impress people.
The point of artistic pursuit has always been to convey meaning from a creator to an audience. That needs to be the primary motivation in your art: to share meaning with others. That being said, if you’re very good at sharing meaning with others through your art, odds are good that you are impressing people. Besides, people are more likely to be impressed by someone doing their best at honest work than by someone just trying hard to impress them.
The other thing about “proving yourself” as an artist of any kind (especially if you are looking to turn it into paying work), is showing your work. When I talk about showing your work, I don’t necessarily mean acquiring a fancy degree or special skills for your resume (although those certainly don’t hurt). I mean putting your money where your mouth is. If you want to be a singer, you’re going to have to sing for an audience at some point. If you want to be a painter, you have to show someone your paintings. It’s okay if it isn’t perfect. Once more, with feeling: It’s okay if it isn’t perfect.
People who create on the highest levels of their fields tend to fall into one of several categories: talented people who have worked extremely hard to build their skills and bodies of work, genuine prodigies that have worked hard to build their skills and bodies of work, and talented people who worked hard and got extremely lucky. Since you cannot control luck, the true common denominator is hard work. Every famous musician, from Brahms to the Beatles, had to work hard to get where they were. Same goes for painters, playwrights, actors, novelists, chefs, dancers, sculptors, rappers, and every other kind of artist. Here’s the thing about those people that you should remember:
Even the most acclaimed and talented people had to start somewhere, too.