Exercise 6.11. Consider the following modified version of Problem A1: V ∗ (x (0)) = sup {x(t+1)}∞ t=0 X∞ t=0 βt U(x (t), x (t + 1)) subject to x (t + 1) ∈ G(x (t), t), for all t ≥ 0 x (0) given, where the main difference from Problem A1 is that the constraint correspondence is timevarying. Suppose that x (t) ∈ X ⊂ RK + for all t, that U : R2K + → RK + is continuously differentiable, concave and strictly increasing an its first K arguments, and that the correspondence G : RK+1 + ⇒ RK + is continuous and convex-valued. Show that a sequence {x∗ (t + 1)}∞ t=0 with x∗ (t + 1) ∈IntG(x∗ (t), t), t = 0, 1,..., is an optimal solution to this problem given x (0), if it satisfies (6.21) and (6.25).