Pro-algebraic and differential algebraic group structures on affine spaces

We prove the following result, solving a problem raised in an article by A. Buium and Ph. J. Cassidy, to appear in the collected works of E. Kolchin. If G is a differential algebraic group whose underlying set is some affine n-space, then G is unipotent. A key result, possibly of independent interest, concerns infinite-dimensional group schemes: a group scheme whose underlying scheme is Spec (K[X₁, X₂,...]) (K an algebraically closed field of characteristic 0) is a projective limit of unipotent algebraic groups.