Suppose that £ is a compact subset of R. If for every x ∈ E there exist a nonnegative function f = fx and an r = r(x) > 0 such that f is C∞ on R, f(t) = 1 for t ∈ (x - r, x + r), and f(t) = 0 for t ∉ (x - 2r, x + 2r), prove that there exist a differentiable function f, a nonzero constant M, and a bounded, open set V which contains E such that 1 < f(x) < M for all x ∈ E and f(x) = 0 for x ∉ V.