Let A A be a densely defined closed (linear) operator, and { A α } \{ {A_\alpha }\} , { B α } \{ {B_\alpha }\} be two nets of bounded operators on a Banach space X X such that | | A α | | = O ( 1 ) , A α A ⊂ A A α , | | A A α | | = o ( 1 ) ||{A_\alpha }|| = O(1),{A_\alpha }A \subset A{A_\alpha },||A{A_\alpha }|| = o(1) , and B α A ⊂ A B α = I − A α {B_\alpha }A \subset A{B_\alpha } = I - {A_\alpha } . Denote the domain, range, and null space of an operator T T by D ( T ) D(T) , R ( T ) R(T) , and N ( T ) N(T) , respectively, and let P ( resp . B ) P(\operatorname {resp} .B) be the operator defined by P x = lim α A α x ( r e s p . B y = lim α B α y ) Px = {\lim _\alpha }{A_\alpha }x(resp. By = {\lim _\alpha }{B_\alpha }y) for all those x ∈ X ( resp . y ∈ R ( A ) ¯ ) x \in X(\operatorname {resp} .y \in \overline {R(A)} ) for which the limit exists. It is shown in a previous paper that D ( P ) = N ( A ) ⊕ R ( A ) ¯ , R ( P ) = N ( A ) , D ( B ) = A ( D ( A ) ∩ R ( A ) ¯ ) , R ( B ) = D ( A ) ∩ R ( A ) ¯ D(P) = N(A) \oplus \overline {R(A)} ,R(P) = N(A),D(B) = A(D(A) \cap \overline {R(A)} ),R(B) = D(A) \cap \overline {R(A)} , and that B B sends each y ∈ D ( B ) y \in D(B) to the unique solution of A x = y in R ( A ) ¯ Ax = y{\text { in }}\overline {R(A)} . In this paper, we prove that D ( P ) = X D(P) = X and | | A α − P | | → 0 ||{A_\alpha } - P|| \to 0 if and only if | | B α | D ( B ) − B | | → 0 ||{B_\alpha }|D(B) - B|| \to 0 , if and only if | | B α | D ( B ) | | = O ( 1 ) ||{B_\alpha }|D(B)|| = O(1) , if and only if R ( A ) R(A) is closed. Moreover, when X X is a Grothendieck space with the Dunford-Pettis property, all these conditions are equivalent to the mere condition that D ( P ) = X D(P) = X . The general result is then used to deduce uniform ergodic theorems for n n -times integrated semigroups, ( Y ) (Y) -semigroups, and cosine operator functions.