Cauchy sequences

One of the problems with deciding if a sequence is convergent is that you need to have a limit before you can test the definition.

Bernard Bolzano was the first to spot a way round this problem by using an idea first introduced by the French mathematician Augustin Louis Cauchy (1789 to 1857).

Definition

A sequence is called a Cauchy sequence if the terms of the sequence eventually all become arbitrarily close to one another.
That is, given ε > 0 there exists N such that if m, n > N then |a_m- a_n| < ε.

Remarks

1. Note that this definition does not mention a limit and so can be checked from knowledge about the sequence.

2. It is not enough to have each term "close" to the next one. (|a_m- a_m+1| < ε. For example, the divergent sequence of partial sums of the harmonic series (see this earlier example) does satisfy this property, but not the condition for a Cauchy sequence.

3. We will see (shortly) that Cauchy sequences are the same as convergent sequences for sequences in R. However, we will see later that when we introduce the idea of convergent in a more general context Cauchy sequences and convergent sequences may be different.

4. Cantor (1845 to 1918) used the idea of a Cauchy sequence of rationals to give a constructive definition of the Real numbers independent of the use of Dedekind Sections.
   

Some properties of Cauchy sequences

1. Any Cauchy sequence is bounded.

   Proof
   (When we introduce Cauchy sequences in a more general context later, this result will still hold.)
   The proof is essentially the same as the corresponding result for convergent sequences.
   
   
   

2. Any convergent sequence is a Cauchy sequence.

   Proof
   If (a_n)→ α then given ε > 0 choose N so that if n > N we have |a_n- α| < ε. Then if m, n > N we have |a_m- a_n| = |(a_m- α) - (a_m- α)| ≤ |a_m- α| + |a_m- α| < 2ε.
   
   
   

3. The Main Result about Cauchy sequences

   A Real Cauchy sequence is convergent.

   Proof
   Since the sequence is bounded it has a convergent subsequence with limit α.
   Claim:
   This α is the limit of the Cauchy sequence.
   Proof of that:
   Given ε > 0 go far enough down the subsequence that a term a_n of the subsequence is within ε of α. Provided we are far enough down the Cauchy sequence any a_m will be within ε of this a_n and hence within 2ε of α.
   
   
   

Remarks

1. The fact that in R Cauchy sequences are the same as convergent sequences is sometimes called the Cauchy criterion for convergence.

2. The use of the Completeness Axiom to prove the last result is crucial. For example, let (a_n) be a sequence of rational numbers converging to an irrational.
   [e.g. (1, 1.4, 1.41, 1.414, ... )→ √2 ]
   Then since (a_n) is a convergent sequence in R it is a Cauchy sequence in R and hence also a Cauchy sequence in Q. But it has no limit in Q.

3. In fact one can formulate the Completeness axiom in terms of Cauchy sequences.
   Here are some equivalent formulations of the axiom

   III Every subset of R which is bounded above has a least upper bound.

   III* In R every bounded monotonic sequence is convergent.

   III** In R every Cauchy sequence is convergent.

   We will see later that the formulation III** is a useful way of generalising the idea of completeness to structures which are more general than ordered fields.
   

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JOC September 2001