Theorem 3.23 is a very simple one: it says that if a series converges then the limit of the terms of the sequence is zero. However Rudin's way of justifying this fact doesn't seem valid to me. He uses the following logic:(adsbygoogle = window.adsbygoogle || []).push({});

A series converges if and only if the sequence of partial sums is cauchy meaning that for all ε > 0 there is an integer N s.t. for all n,m > N and n <= m the sum of the terms of the sequence from a_n to a_m is less than ε.

Rudin says that the case where n = m proves this theorem. However when n = m the only thing the cauchy criterion states is that the distance from a_n to a_n approaches zero. It does not actually say that thevalueof a_n approaches zero.

To prove this we need the case where n = m - 1.

Then the difference between the two partial sums is a_m and therefore a_m approaches zero.

Am I wrong?

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Rudin Theorem 3.23

Loading...

Similar Threads for Rudin Theorem | Date |
---|---|

I Heine-Borel Theorem shouldn't work for open intervals? | Jun 26, 2016 |

Rudin's Principles Theorem 1.11 (supremum, infimum) | Feb 1, 2015 |

Rudin Theorem 1.21. How does he get The identity ? | Nov 8, 2013 |

Baby Rudin Theorem 1.11 | Oct 20, 2013 |

Rudin Proof of Liouville Theorem (Complex A.) | Apr 20, 2013 |

**Physics Forums - The Fusion of Science and Community**