What should i do to prove to employers to hire me as a developer?
Usually employers will take a chance on some one who is inexperienced if they can prove they are a passionates programmer, show that that you have some initiative. Work on your own projects, or at least join discussion groups in your desired field such as meetups.
My short story:
I graduated college w/ a degree in EE and after working for a defense company for about a year, realized I hated it. Web dev/design was my hobby and I had hacked together a lot of random projects, but never had any "official industry experience". On my resume I had listed a lot of side projects I had. I applied to a few places and a few called me in for an interview and also a short test of what I knew. This is where you will get a chance to try and prove yourself. I was nervous as heck, but thankfully, they gave me a chance and the rest was history.
I don't think I would have been given the chance had I not been doing those things on the side. The other nice side effect was that because I actively worked on projects, I actually knew what I was doing and for some reason that surprised a lot of employers(?).
I think the one thing I took away from that was that if you're passionate about learning and improving, you'll be better off than the majority of people just there looking for a job. "3+ years" exp. or whatever had a different meaning for me after that.