Quick Background

Since learning basic html 5 years ago, I've become completely obsessed with the technology, the logic, and the thrill of solving problems involved with building websites. I am still stuck at a thoroughly non-programming type job, but would really like to move into the field of web programming/design.

I have no educational background in the field (was trained as a fine artist and tutor), but in the past few years have progressed fully self-taught (and self-motivated) from html to css to php, mysql, jquery, and am now building rich web applications.

The Question

How can I prove to a company that even though I have no education, I have a passion to learn whatever is thrown my way? ...That essentially I would come at every issue with not only knowledge, but with a passionate desire to solve it, whether that means tackling a new language or debugging code for hours at a time?

p.s. Sorry for the stupid title.