So I am trying to prove that a sequence is cauchy if and only if it is convergent using only the Bolzano Weierstrass therom for sequences
[Any bounded sequence must have at least one convergent subsequence]
So far, all I have gotten in the forward direction (cauchy => convergent) is that a cauchy sequence is bounded, and hence by the Bolzano Weierstrass theorem, there must be at least ONE convergent subsequence. I have a theorem in the book which states:
[A sequence converges to A iff each of its subsequences converge to A.]
Is this the proper theorem to use in my problem and if so how do I show that ALL of the subsequences converge, or even that more than one subsequence exists at all?
Also in the backwards direction (convergent => cauchy), I am completely at a loss. Any help would be greatly appreciated.
Thanks!
[QUOTE=dannyboycurtis;380174]So I am trying to prove that a sequence is cauchy if and only if it is convergent using only the Bolzano Weierstrass therom for sequences
[Any bounded sequence must have at least one convergent subsequence]
So far, all I have gotten in the forward direction (cauchy => convergent) is that a cauchy sequence is bounded, and hence by the Bolzano Weierstrass theorem, there must be at least ONE convergent subsequence. I have a theorem in the book which states:
[A sequence converges to A iff each of its subsequences converge to A.]
Is this the proper theorem to use in my problem and if so how do I show that ALL of the subsequences converge, or even that more than one subsequence exists at all?[quote]
Let be your sequence and be the subsequence that converges to A. Given any there exist such that if and is in the convergent subsequence then . Also, because is a Cauchy sequence, there exist such that if m and n are both larger than , [tex]|a_m- a_n|< \epsilon/2.
Now, let N be the larger of and so that if n> N, both are true. Since is an infinite sequence, there certainly exist also. Then, for n> N, and both are less than .
Plato showed this. And, by the way, the proof in this direction does NOT require "Bolzano Weierstrasse" or any form of "completeness". For example, in the rational numbers Cauchy sequences are not necessarily convergent, but any convergent sequence is a Cauchy sequence.Also in the backwards direction (convergent => cauchy), I am completely at a loss. Any help would be greatly appreciated.
Thanks!