An elementary proof of the Briancon-Skoda theorem

We give a new elementary proof of the Brian\c{c}on-Skoda theorem, which states that for an $m$-generated ideal $\mathfrak{a}$ in the ring of germs of analytic functions at $0\in \C^n$, the $\nu$:th power of its integral closure is contained in $\mathfrak{a}$, where $\nu = \min(m,n)$.