Nonlinear Programming

Part 1 (if): Assume that Z is closed. We must show that if A is closed for all k → x x , k → y y , where ( k A ∈ ) k y x , then ( ) A ∈ y x . By the definition of Z being closed, we know that all points arbitrarily close to Z are in Z. Let k → x x , k → y y , and ( k A ∈ ) k y x . Now, for any ε > 0, there exists an N such that for all k ≥ N we have || || k ε − < x x , || || k ε − < y y which implies that ( ) , x y is arbitrarily close to Z, so ( ) , x y ∈ Z and ( ) A ∈ y x . Thus, A is closed.