Cauchy Sequences

In his construction of real numbers as (equivalence classes of special kinds of) sequences of rationals, Cauchy wanted a way to identify convergent sequences without having to specify the limit in advance. Noting that a sequence converges if and only if its values get arbitrarily close to the limiting value, and thus arbitrarily close to each other, he idenitified the sequences now named for him:

Definition 4   A sequence $a:\ensuremath{\mathbb{N} }\to \ensuremath{\mathbb{R} }$ is a Cauchy sequence if for every $\epsilon$ there is an M such that if n>m>M then $\vert a_n-a_m\vert<\epsilon$.

We can see quickly that any convergent sequence is Cauchy: given $\epsilon$ choose M so that if n>M then $\vert a_n-L\vert<\frac{\epsilon}{2}$. If both n and m are larger than M then

$$\vert a_n-a_m\vert\leq \vert a_n-L\vert+\vert L-a_m\vert<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon$$

The other direction, however, is far from obvious.

Theorem 6   Every Cauchy sequence converges.

PROOF:

Let a be a Cauchy sequence. Then $\{a_n\vert n\in\ensuremath{\mathbb{N} }\}$ is bounded (mimic the proof that convergent sequences are bounded, but use a_M+1 instead of L). There is also a subsequence a_f(n) of a which is monotone, and thus must converge, say to L. Now given $\epsilon$ let M be large enough that for f(n), m>M we get both $\vert a_{f(n)}-L\vert<\epsilon/2$ and $\vert a_m-a_{f(n)}\vert<\epsilon/2$. Combining thes we see that for m>M we get $\vert a_m-L\vert<\epsilon$.