# proof of Silverman-Toeplitz theorem

First, we shall show that the series ${\sum}_{n=0}^{\mathrm{\infty}}{a}_{mn}{z}_{n}$ converges. Since the sequence $\{{z}_{n}\}$ converges, it must be bounded^{} in absolute value^{} — there must exist a constant $K>0$
such that $|{z}_{n}|\le K$ for all $n$. Hence, $|{a}_{mn}{z}_{n}|\le K|{a}_{mn}|.$ Summing this gives

$$\sum _{n=0}^{\mathrm{\infty}}|{a}_{mn}{z}_{n}|\le K\sum _{n=0}^{\mathrm{\infty}}|{a}_{mn}|\le KB.$$ |

Hence, the series ${\sum}_{n=0}^{\mathrm{\infty}}{a}_{mn}{z}_{n}$ is absolutely convergent which, in turn, implies that it converges.

Let $z$ denote the limit of the sequence $\{{z}_{n}\}$ as $n\to \mathrm{\infty}$. Then $|z|\le K$. We need to show that, for every $\u03f5>0$, there exists an integer $M$ such that

$$ |

whenever $m>M$.

Since the sequence $\{{z}_{n}\}$ converges, there must exist an integer ${n}_{1}$ such that $$ whenever $n>{n}_{1}$.

By condition 3, there must exist constants ${m}_{o},{m}_{1},\mathrm{\dots},{m}_{{n}_{1}}$ such that

$$ | (1) |

Choose ${m}^{\prime}=\mathrm{max}\{{m}_{0},{m}_{1},\mathrm{\dots},{m}_{{n}_{1}}\}.$ Then

$$ | (2) |

when $m>{m}^{\prime}.$

By condition 2, there exists a constant ${m}^{\prime \prime}$ such that

$$ |

whenever $m>{m}^{\prime \prime}$. By (1),

$$ |

when $m>{m}^{\prime}$. Hence, if $m>{m}^{\prime}$ and $m>{m}^{\prime \prime}$, we have

$$ | (3) |

and

$$ | (4) |

By the triangle inequality^{} and (2), (3), (4) it follows that

$$\left|\sum _{n=0}^{\mathrm{\infty}}{a}_{mn}{z}_{n}-z\right|\le \left|\sum _{n=0}^{{n}_{1}}{a}_{mn}{z}_{n}\right|+\left|\sum _{n={n}_{1}+1}^{\mathrm{\infty}}{a}_{mn}({z}_{n}-z)\right|+|z|\left|\sum _{n={n}_{1}+1}^{\mathrm{\infty}}{a}_{mn}-1\right|$$ |

$$ |

Title | proof of Silverman-Toeplitz theorem |
---|---|

Canonical name | ProofOfSilvermanToeplitzTheorem |

Date of creation | 2013-03-22 14:51:35 |

Last modified on | 2013-03-22 14:51:35 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 27 |

Author | rspuzio (6075) |

Entry type | Proof |

Classification | msc 40B05 |