Silverman-Toeplitz theorem

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

1. ${\sum }_{n=0}^{\mathrm{\infty }}|a_{mn}|\le B$ for all $m=0,1,2,\mathrm{\dots }$

2. ${lim}_{m\to \mathrm{\infty }}{\sum }_{n=0}^{\mathrm{\infty }}{a}_{mn}=1$

3. For every $n=0,1,2,\mathrm{\dots }$, it is the case that ${lim}_{m\to \mathrm{\infty }}{a}_{mn}=0$

Then, if the sequence $\{z_n\}$ converges, the series ${\sum }_{n=0}^{\mathrm{\infty }}{a}_{mn}{z}_n$ converges and

$$\underset{n\to \mathrm{\infty }}{lim}{z}_n=\underset{m\to \mathrm{\infty }}{lim}\sum _{n=0}^{\mathrm{\infty }}{a}_{mn}{z}_n$$