  We present a proof that the hyperbolic plane cannot be isometrically immersed
in Euclidean $3$-space by a $C^\infty$ map. Ideas from many topics in
(essentially) undergraduate mathematics are applied; the use of moving frames
and connection forms to express the geometry simplifies the outline of the
proof, compared to, say, using coordinate patches and Christoffel symbols. The
key transition is from expressions in terms of the principal directions on the
immersed surface (which give access to the Gaussian curvature) to expressions
in terms of the asymptotic directions (which yield a coordinate system and give
access to surface area).
