We give a simple finitary proof that every Goodstein Sequence G(m) over the
natural numbers must terminate finitely. We note first that if G(p) terminates
finitely then so does G(m) for all natural numbers m<p. We then show that for
every natural number k there is a natural number u such that the (k-1)th term
of G(u) is k^k, and that G(u) terminates finitely.