A non-standard proof of the Briançon-Skoda theorem

Using a tight closure argument in characteristic $p$ and then lifting the argument to characteristic zero with the aid of ultraproducts, I present an elementary proof of the Briançon-Skoda Theorem: for an $m$-generated ideal $\mathfrak{a}$ of ${\mathbb C}[[{X_1,\dots,X_n}]]$, the $m$-th power of its integral closure is contained in $\mathfrak{a}$. It is well-known that as a corollary, one gets a solution to the following classical problem. Let $f$ be a convergent power series in $n$ variables over $\mathbb C$ which vanishes at the origin. Then $f^n$ lies in the ideal generated by the partial derivatives of $f$.