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?

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.
