Why complete space is good? 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.
