Our 'intro to Analysis' professor intruduced compactness of subsets of metric spaces by first introding what he calls 'rijcompact' (sequential compactness), and going on to prove that it is equivalent to compactness.
A subset $V$ of a metric space $X,d$ is sequentialy-compact if every sequence has a convergent subsequence.
In our course-notes, there's a proof that sequential compactness is equivalent to complete and totally bounded. The proof constructs a Cauchy-subsequence of a sequence in a sequentially compact subset of a metric space and concludes that it is convergent because the metric space is complete.
The first part of the proof is rather long. It could be shortened if there was a definition of this property of a subset of a subset of a metric space:
A subset $V$ of a metric space $X,d$ is [INSERT NAME HERE] if every sequence has a Cauchy subsequence.
Is there a name for this property?