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Introduction to Modern Economic Growth
some x (1) ∈ G (x (0)). In view of Assumption 6.1, V ∗ (x (0)) is finite. Moreover,
Assumptions 6.1 and 6.2 also enable us to apply Weierstrass theorem to Problem
A1, thus there exists x ∈Φ (x (0)) attaining V ∗ (x (0)) (see Mathematical Appendix).
A similar reasoning implies that there exists x0 ∈Φ (x (1)) attaining V ∗ (x (1)). Next,
since (x (0) , x0 ) ∈ Φ (x (0)) and V ∗ (x (0)) is the supremum in Problem A1 starting
with x (0), Lemma 6.1 implies
V ∗ (x (0)) ≥ U (x (0) , x (1)) + βV ∗ (x (1)) ,
= U (x (0) , x0 (1)) + βV ∗ (x0 (1)) ,
thus verifying (6.11).
Next, take an arbitrary ε > 0. By (6.10), there exists x0ε = (x (0) , x0ε (1) , x0ε (2) , ...) ∈Φ (x (0))
such that
¯ (x0ε ) ≥ V ∗ (x (0)) − ε.
U
Now since x00ε = (x0ε (1) , x0ε (2) , ...) ∈ Φ (x0ε (1)) and V ∗ (x0ε (1)) is the supremum in
Problem A1 starting with x0ε (1), Lemma 6.1 implies
¯ (x00ε ) ≥ V ∗ (x (0)) − ε
U (x (0) , x0ε (1)) + β U
U (x (0) , x0ε (1)) + βV ∗ (x0ε (1)) ≥ V ∗ (x (0)) − ε,
The last inequality verifies (6.12) since x0ε (1) ∈ G (x (0)) for any ε > 0. This proves