  We prove that any connected proper Dupin hypersurface in $\R^n$ is analytic
algebraic and is an open subset of a connected component of an irreducible
algebraic set. We prove the same result for any connected non-proper Dupin
hypersurface in $\R^n$ that satisfies a certain finiteness condition. Hence any
taut submanifold M in $\R^n$, whose tube $M_\epsilon$ satisfies this finiteness
condition, is analytic algebraic and is a connected component of an irreducible
algebraic set. In particular, we prove that every taut submanifold of dimension
$m \leq 4$ is algebraic.
