I wonder why the complete space is a good space. I know that for a metric space $X$, if any Cauchy sequence $\{x_n\}$ in $X$ converges, then $X$ is called complete metric space. But why such property is good?
Here, 'good' means basically why such property is useful. Also, why such property is desired. Why the 'completion' matters. So yes I need proper motivation for such concept.
Ok, every Cauchy sequence converges, so what?
Just an example. Any contraction on a nonempty complete metric space has a unique fixed point.
We know that $\{x_n\}$ converges in $X$ implies $\{x_n\}$ is Cauchy. If $X$ is complete, you have the converse, so you can assume that Cauchy sequences are convergent (if $X$ is not complete, you have the completion of $X$). Now, being Cauchy is intrinsic to the sequence, that is, you don't need a limit point in order to check that a sequence is Cauchy. That's very useful when you want to prove that a certain sequence converges without having a candidate for limit.
The notion of completeness is mosty inspired on the reasons why we have the real numbers as a basic system, and not just the rationals $\Bbb Q$: the Greeks already found that there are line segments that could not be rational length ($\sqrt{2}$ e.g., the suspected about $\pi$, but could not prove that), but that could be approximated by rationals (e.g. Archimedes' inscribed polygons etc.). But the Greeks could't make the step to completeness at the time: they had an idea that a line segments is complete (every point you "see" must exist somehow, but not of limits : limit process yes, but you couldn't "reach" the limit, Aristotle said there as no actual infinity, just potential infinite... So philosophically they weren't ready for it, as it were.
In the 19th century Cantor (and Dedekind and others) formalised how to construct real numbers as a "completion" of rational numbers (so that they could prove the consistency of the real number system, of which mathematicians already used completeness-like properties: e.g. that all square roots, and roots of polynomials (algebraic numbers) somehow existed. Cantor used rational Cauchy sequences that have no rational limit as a witness to the "holes" in the rationals: numbers that "should" exist, but didn't in the rationals. Dedekind did a similar thing for order ($\sqrt{2}$ as a supremum of a set of rationals that has no supremum in the rationals).
The idea was so useful (because it allows to prove the consistency of $\Bbb R$) that it was generalised to all metric (and uniform spaces) to achieve similar results: to get enough elements to be useful, in a way. We have existence theorems in complete spaces that do not hold in general: Tarski's fixed point theorem, Banach's fixed point theorem, the intermediate value theorem, the fact that bounded continuous functions achieve their max or min, all direct or indirect consequences of completeness.
Another example-family concerns (topological/metric) vector spaces of functions of various types. "Completeness" means that a Cauchy sequence ... described via the metric or other topology ... of a sequence of functions is again in the same space.
So, for example, while pointwise limits of continuous functions on $[a,b]$ can be pretty crazy, uniform pointwise limits (that is, Cauchy sequences with respect to the metric $d(f,g)=\sup_{x\in [a,b]}|f(x)-g(x)|$) are again continuous.
Similar results make $L^p$ spaces interesting/useful, as well as Sobolev spaces, and $C^k$ spaces, and ... :)
(I've forgotten to reiterate the perhaps-obvious point that we'd like to know what kind of limits of what kind of functions are of the same kind...)
