The optimal paper Moebius band

We prove that a smooth embedded paper Moebius band must have aspect ratio greater than $\sqrt(3)$. We also prove that any sequence of smooth embedded paper Moebius bands whose aspect ratio converges to $\sqrt{3}$ must converge, up to isometry, to the triangular Moebius band. These results answer the mimimum aspect ratioquestion discussed by W. Wunderlich in 1962 and the more specific conjecture of B. Halpern and C. Weaver from 1977.