Introduction to Modern Economic Growth number) By the converse argument, g (x) ≤ f (x) + kg − f k for any x ∈ X, (T g) (x) ≤ T [f + kg − f k] (x) for any x ∈ X, (T g) (x) ≤ (T f ) (x) + β kg − f k for any x ∈ X Combining the last two inequalities implies kT f − T gk kf gk , Ô proving that T is a contraction We will see that Blackwell’s sufficient conditions are straightforward to check in many economic applications, including the models of optimal or equilibrium growth 6.4 Proofs of the Main Dynamic Programming Theorems* We now prove Theorems 6.1-6.6 We start with a straightforward lemma, which will be useful in these proofs For a feasible infinite sequence x = (x (0) , x (1) , ) ∈ Φ(x (0)) starting at x (0), let ¯ (x) ≡ U ∞ X β t U (x (t) , x (t + 1)) t=0 be the value of choosing this potentially non-optimal infinite feasible sequence In ¯ (x) exists and is finite The next lemma shows that U ¯ (x) view of Assumption 6.1, U can be separated into two parts, the current return and the continuation return Lemma 6.1 Suppose that Assumption 6.1 holds Then for any x (0) ∈ X and any x ∈ Φ(x (0)), we have that ¯ (x) = U(x (0) , x (1)) + β U(x ¯ 0) U where x0 = (x (1) , x (2) , ) 272