User:Gmalivuk

$f_{n}(k)={\begin{cases}1,&{\text{if }}k=n\\0,&{\text{if }}k\neq n\end{cases}}$

$f(k)=\lim _{n\to \infty }f_{n}(k)=0$

$a_{n}=\sum _{i=1}^{\infty }f_{n}(i)=1$

$\lim _{n\to \infty }\sum _{i=1}^{\infty }f_{n}(i)=\lim _{n\to \infty }a_{n}=1$

$\sum _{i=1}^{\infty }\lim _{n\to \infty }f_{n}(i)=\sum _{i=1}^{\infty }f(i)=0$