Completeness proof? First of all, this is not a question about a specific problem, but more about a general technique. When I face a problem such as "show that a metric space $(M,d)$ is complete", the first thing I do is to say: if a metric space is complete, then every Cauchy sequence $(x_n)$ where $x_n\in M$ for all $n$ converges in $M$. From there, I am clueless as to how to proceed: are we allowed to assume that $(x_n)$ at least converges to something that may or may not be in $M$?

Classic three-steps proof once you take a Cauchy sequence in your space:  
(i) find a candidate for your limit (limit something that sounds the right possible limit for the third step),  
(ii) show it belongs to the space (definition of being in the space),  
(iii) show your sequence actually converges to it (definition of cenvergence).
