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).
- 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 |am- an| < ε.
- 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. (|am- am+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.
(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.
If (an)→ α then given ε > 0 choose N so that if n > N we have |an- α| < ε. Then if m, n > N we have |am- an| = |(am- α) - (am- α)| ≤ |am- α| + |am- α| < 2ε.
- The Main Result about Cauchy sequences
A Real Cauchy sequence is convergent.
Since the sequence is bounded it has a convergent subsequence with limit α.
This α is the limit of the Cauchy sequence.
Proof of that:
Given ε > 0 go far enough down the subsequence that a term an of the subsequence is within ε of α. Provided we are far enough down the Cauchy sequence any am will be within ε of this an and hence within 2ε of α.
- 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 (an) be a sequence of rational numbers converging to an irrational.
[e.g. (1, 1.4, 1.41, 1.414, ... )→ √2 ]
Then since (an) 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.
JOC September 2001