I was recently reviewing some notes regarding compactness, in which the sequential definition is given i.e. "$A$ is compact if any sequence in $A$ has a subsequence which converges to a limit in $A$. So in this instance we are only concerned with sequences in $A$ which actually converge.
I can recall one of my analysis lectures where someone mentioned thinking of infinity as a point on the real line (for some unrelated topic), and this got me wondering how compactness/completeness would be effected if we decided to include infinity in the reals as if it was a number (or for any metric space).
From what I've determined, all of the sequences which diverge to infinity are now considered convergent, and so the previously compact sets are now no longer compact since they do not include the limits of every convergent sequence, and the only compact space could be the whole space itself correct? (Unless we can place infinity in some arbitrary point not on the boundary).
Also, In trying to work with this idea, I am struggling to come to terms with how this would effect Cauchy sequences. Given that the previously "divergent" sequences now converge, if our triangle inequality still holds wouldn't they now be Cauchy since they get closer to infinity as they grow larger? Shouldn't we have to change our notion of distance in this new space to avoid a contradiction?