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 $M$?

In practice, you often construct your limit object. It can be by using the completeness of a well-known space, generally $\mathbb{R}$ itsel. You probably know the following examples but the principle is very general. 
Consider $C_b$ the space of continuous and bounded functions from $\mathbb{R}$ to itself, with the norm of uniform convergence $\lVert \cdot \lVert_\infty$. Given a Cauchy sequence $f_n$, you know then that $f_n(x)$ is a real Cauchy sequence for all $x$ and it allows you to defined $f(x)$ as the limit (using that $\mathbb{R}$ is complete) of this sequence. Thus, you have define a function $f$ and then can show that your Cauchy sequence converges to that limit. 
Another example that doesn't use the completeness of $\mathbb{R}$. To show that the space $L^p$ of measurable functions with finite $L^p$-norm is complete, you can proceed the following way. Given a Cauchy sequence for the $L^p$ norm, you show that a subsequence converges almost surely, which defines clearly your limit object, and then show that the whole sequence converges.
Very often, you will have to follow that plan: construct your limit, and show that your
