Calculus III: Open sets proof help

I need to prove that the following is an open subset of R^2:

$$\left\{(x,y)\in$$R$$^{2}$$|$$\sqrt{x^2+y^2}$$<1}

I think the substition r=min{sqrt[x^2+y^2],1-sqrt[x^2+y^2]} works, but I'm stuck on how to take it from that to showing that the distance between X0 and X1 is less that r, and more importantly, proving that this means that the subset is open. Any help would be must appreciated.