  We give a short proof of the following fact. Let $\Sigma$ be a connected,
finitely connected, noncompact manifold without boundary. If $g$ is a complete
Riemannian metric on $\Sigma$ whose Gaussian curvature $K$ is nonnegative at
infinity, then $K$ must be integrable. In particular, we obtain a new short
proof of the fact that if $\Sigma$ admits a complete metric whose Gaussian
curvature is nonnegative and positive at one point, then $\Sigma$ is
diffeomorphic to $\mathbb{R}^2$.
