  We present a simple, computation free and geometrical proof of the following
classical result: for a diffeomorphism of a manifold, any compact submanifold
which is invariant and normally hyperbolic persists under small perturbations
of the diffeomorphism. The persistence of a Lipschitz invariant submanifold
follows from an application of the Schauder fixed point theorem to a graph
transform, while smoothness and uniqueness of the invariant submanifold are
obtained through geometrical arguments. Moreover, our proof provides a new
result on persistence and regularity of "topologically" normally hyperbolic
submanifolds, but without any uniqueness statement.
