I'm reading some stuff on completeness and I can't work out if it's just so long since Analysis I or whether because they never told us but I don't see why Cauchy Sequences are so important when it comes to completeness. Surely you could define it so that any sequence that converges must have a limit in the set since any convergent sequence is necessarily Cauchy, or is there something more at work here?

See, convergence of sequences is written in terms of a known point , if it gets arbritrarily close to and stays there (er... roughly). It is the natural definition of tending to a limit, and certainly should feel so after metric spaces.

But mathematics is all about maximal results with minimal hypotheses- and it makes sense to search for a weaker definition of limit. A cauchy sequence is one such weakened definition, heuristically a cauchy sequence is a sequence that converges with respect to itself, a train that comes to a stop; but that need not do so at a station - there need be no point for it to tend to, a cauchy sequence simply tends.

The reason why the definition seems pointless to begin with is that most examples of incomplete spaces you will be shown are subspaces of larger complete spaces and it is tempting to view them thus*, but the abstraction can be useful: and there are familiar seeming incomplete spaces with rather unfamiliar completions (try the P-adic numbers on for size...).

All in all, like much in mathematics, it is an abstraction that is trivial for familiar examples, a weak hypothesis that comes up equivalent to standard stuff for "nice" things: but if you go deep enough it draws some pretty important distinctions.

*(indeed every such space can abstractly be viewed as a subspace of its completion- see Sutherland's rather fab bit on the construction of the reals in the metric spaces core text for an example)