User:Gmalivuk

$f_n(k)=\{\begin{array}{ll}1,& \text{if }k=n\\ 0,& \text{if }k\ne n\end{array}$

$f(k)=\underset{n\to \mathrm{\infty }}{\lim }{f}_n(k)=0$

$a_n=\sum _{i=1}^{\mathrm{\infty }}{f}_n(i)=1$

$\underset{n\to \mathrm{\infty }}{\lim }\sum _{i=1}^{\mathrm{\infty }}{f}_n(i)=\underset{n\to \mathrm{\infty }}{\lim }{a}_n=1$

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