Silverman-Toeplitz theorem

Let $\{a_{mn}\}$ be a double sequence of complex numbers and let $B$ be a positive real number such that:

1.

$\sum_{n=0}^{\infty}|a_{mn}|\leq B$ for all $m=0,1,2,\ldots$
2.

$\lim_{m\to\infty}\sum_{n=0}^{\infty}a_{mn}=1$
3.

For every $n=0,1,2,\ldots$ , it is the case that $\lim_{m\to\infty}a_{mn}=0$

Then, if the sequence $\{z_{n}\}$ converges, the series $\sum_{n=0}^{\infty}a_{mn}z_{n}$ converges and

$\lim_{n\to\infty}z_{n}=\lim_{m\to\infty}\sum_{n=0}^{\infty}a_{mn}z_{n}$