User:Gmalivuk


 * $$f_n(k) = \begin{cases}

1, & \text{if } k = n \\ 0, & \text{if } k \ne n \end{cases} $$


 * $$f(k) = \lim_{n\to\infty}f_n(k) = 0$$


 * $$a_n = \sum_{i=1}^\infty f_n(i) = 1$$


 * $$\lim_{n\to\infty}\sum_{i=1}^\infty f_n(i) = \lim_{n\to\infty}a_n = 1$$


 * $$\sum_{i=1}^\infty \lim_{n\to\infty}f_n(i) = \sum_{i=1}^\infty f(i) = 0$$