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

*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*| < 2*ε*.

**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 2*ε*of*α*.

**Remarks**- The fact that in
**R**Cauchy sequences are the same as convergent sequences is sometimes called the*Cauchy criterion for convergence*. - The use of the Completeness Axiom to prove the last result is crucial. For example, let (
*a*) be a sequence of rational numbers converging to an irrational._{n}

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

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**