For p=2 and tame level N=1 we prove that the map from the (Coleman-Mazur)
Eigencurve to weight space satisfies the valuative criterion of properness.
More informally, we show that the Eigencurve has no "holes"; given a punctured
disc of finite slope overconvergent eigenforms over weight space, the center
can be "filled in" with a finite slope overconvergent eigenform
