Usefulness of Cauchy sequences

Question

I took two courses in single- and multivariable calculus. Both of which dealt with Cauchy sequences. My question is now, why is the property of being a Cauchy sequence useful? I know that it is used to define complete (metric) spaces, but is there any way in which these sequences are used?

Answer 1

Cauchy sequences have several important applications:

• Since the Cauchy sequences in a metric space admit a natural equivalence relation (namely $d(x_n,\,y_n)\to0$, with $d$ the space's metric), the equivalence classes provide a metric-completion of the original space. If a metric space is metric-complete, its Cauchy sequences converge; otherwise, we can only guarantee the converse. For example, Cauchy sequences in $\Bbb Q$ need not converge to some limit in $\Bbb Q$, but they will have a limit in its metric completion $\Bbb R$, as will all Cauchy sequences in $\Bbb R$ itself. @JoséCarlosSantos already discussed why metric completeness is useful when it occurs. Better still, if you change the norm used to define Cauchy sequences, you get a different metric completion (see here and here).
• Metric completion also holds in Banach spaces and Hilbert spaces, important classes of vector space that allow us to add infinitely many vectors (as long as the partial sums comprise a Cauchy sequence). Whereas the usual definition of a basis, named for Hamel, requires finite linear combinations thereof to span the space, in the above special cases we can instead use a Schauder basis, for which countably infinite combinations are legal. There's even an uncountable variant, using integrals. These are both vital for the Hilbert spaces used in quantum mechanics.
• On a related note, important theorems on such spaces may be proved with Cauchy sequences, such as the Banach fixed-point theorem.

Answer 2

It is useful because it allows us to prove that a sequence converges even without knowing what its limit is.

Consider, for instance, the statement “Every absolutely convergent series converges.” This is proved by proving that, given an absolutely convergent series $\sum_{n=0}^\infty a_n$, the sequence $\left(\sum_{n=0}^Na_n\right)_{N\in\mathbb Z_+}$ is a Cauchy sequence. And so we do not have to know what is its sum in order to prove that it converges.