I'm having trouble with this problem:

Suppose X is a paracompact topological space. Let U be an open set of X\times [0,\infty) which contains X\times\{0\}. Show that there is a map f\colon X\rightarrow (0,\infty) such that y\leq f(x) implies (x,y)\in U.

Note that all my paracompact spaces are assumed to be Hausdorff.

I know that paracompact spaces are in particular normal, and I tried to use an Urysohn function, but I couldn't find one with the right property.

I also attempted a partition of unity, but the codomain of f is not supposed to contain 0, which throws me off.