Generalized Ordinary Differential Equations in Abstract Spaces and Applications. Группа авторов

, there is a division for which

      for every and , .

Take . Because the sequence converges pointwisely to , we have and also , for . Thus, for every , there is such that, whenever , we have and for

      Take an arbitrary . Then, either for some , or for some . In the former case, The other case yields

      Then, and, therefore, uniformly on .

      

      Lemma 1.14: Let be a sequence in . The following assertions hold:

      1 if the sequence of functions converges uniformly to as on , then , for , and , for ;

      2 if the sequence of functions converges pointwisely to as on and , for , and , for , where , then the sequence converges uniformly to as .

      Proof. We start by proving . By hypothesis, the sequence converges uniformly to . Then, Moore–Osgood theorem (see, e.g., [19]) implies

      Therefore, , for . In a similar way, one can show that , for every .

      Now, we prove . It suffices to show that