Cauchy Sequences
A Cauchy sequence \(\{a_n\}_{n=1}^\infty\) is one which has the following property:
\[{}\forall \epsilon > 0 \ \exists N \in {\mathbb N} \text{ such that }{}\]
\[{}n,m > N \Rightarrow |a_n - a_m | < \epsilon.{}\]
In other words, for any threshold \(\epsilon\), there is a point beyond which all terms in the tail of the sequence are \(\epsilon\)-close to each other. This is closely related to the definition of convergence but distinct. It's generally easier to show a sequence is Cauchy than it is to show that it converges because the definition of “Cauchyness” does not make reference to any limit.
Warning
It is extremely important to observe that all terms in the tail of a Cauchy sequence are close to each other. This is a much stronger condition than merely saying that adjacent terms are close to one another. This distinction must be kept in mind at all times.
It's not so difficult to prove that all convergent sequences are Cauchy.
Theorem 1 (Convergent Sequences are Cauchy)
If \(\{a_n\}_{n=1}^\infty\) is a convergent sequence of real numbers, then it must be a Cauchy sequence.
Proof
The really remarkable thing is that the converse is true:
Theorem 2 (Cauchy Sequences Converge)
If \(\{a_n\}_{n=1}^\infty\) is a Cauchy sequence of real numbers, then it must be convergent.
The proof of this theorem, like Bolzano-Weierstrass, is a bit involved. The first step is the following intermediate result, which is often useful in its own right.
Proposition
If \(\{a_n\}_{n=1}^\infty\) is Cauchy, then it must be bounded.
Proof
Exercise
- Prove directly from the definitions that a subsequence of a Cauchy sequence is necessarily also Cauchy.