Differential Geometry: Gaussian curvature question

Hi, I was wondering if anyone could help me - given the closed unit disc, with its boundary circle "glued" to the plane, I'm trying to prove that it is impossible to put a Riemannian metric on this space such that the Gaussian curvature is always non-negative, but I'm not really sure how to go about it. From the Gauss-Bonnet theorem, we have that the total curvature of the space is 2(pi), but that doesn't seem to help much...

(I think part of the problem is that I'm not really convinced that what I am trying to prove is true, even with the standard metric, but I've been assured that it is!)

Thanks very much for your help,

Jonathan.