Infinite Intersections of open subschemes and the Hilbert scheme of points

We study infinite intersections of open subschemes and the corresponding intersection of Hilbert schemes. If $\{U_i\}$ is the collection of open subschemes of a variety $X$ containing a fixed point $P$, then we show that the Hilbert functor of flat and finite families on the spectrum of the local ring of $P$ is given by the intersection of ${\text{Cal}H}_i$, where ${\text{Cal}H}_i$ is the Hilbert functor of flat and finite families on $U_i$. In particular we show that the Hilbert functor of flat and finite families on the spectrum of the local ring of $P$ is representable by a scheme.