I'm reading the discussion about convergent sequences defined on a metric space on Simmons "Introduction to Topology and Modern Analysis".
The author explains why trivially a convergent sequence
is a Cauchy sequence and shows how because the convergence of a sequence depends not only of the sequence itself but of the nature of the
metric space, a Cauchy sequence in certain metric space may happen to not converge.
Right after that, he says:
Some writers emphasize the distinction between convergent sequences
and Cauchy sequences by calling the latter "intrinsically convergent"
sequences.
The world "intrinsic" means "belonging to or part of the real nature of something/somebody", then isn't saying that Cauchy sequences in a metric space are intrinsically convergent a total mistake ?
Maybe I'm misunderstanding the meaning of the phrase (I'm not an native english speaker), but I'm thinking of an "intrinsical property" as being something opposite to an "embedding property" (for example, a group H is not normal "per se", but with respect to some group that contains it).
Best Answer
“Intrinsically convergent” wouldn’t mean literally convergent, because we have a term for that already, “convergent.” So, as a technical term you shouldn’t take “intrinsic” quite so literally.
Not every Cauchy sequence in the rationals converges in the rationals, but they do in the real numbers. So, while these sequences don’t converge in $\mathbb Q,$ there is a sense in which $\mathbb Q$ is “missing points” to cover these cases, and that they really do converge “some place.”
More generally, given a metric space, $(X,d),$ there is a unique space $(\overline X,d’),$ called the completion of $X,$ which contains $X$ and in which all Cauchy sequences converge. $\overline X$ is in some sense a “natural” extension of $X.$
The completion of the rational numbers is the reals.
For another example, if $$X=\{(x,y)\in \mathbb R^2\mid x^2+y^2<1\}$$ with the usual Euclidean distance, then the completion is identical to $\{(x,y)\mid x^2+y^2\leq 1\}.$ The Cauchy sequences in $X$ that don’t converge are instead intrinsically converging to the points on the boundary.
So, while a Cauchy sequence in $X$ does not converge, there is a single “natural” extension of $X$ in which all Cauchy do converge.
Essentially, a metric space where some Cauchy sequences do not converge can be thought of as “missing points” in the same way the rationals are missing the irrational reals, or $X$ is missing the boundary.
These points are “intrinsically” there, in that they exist in a natural and unique extension of the incomplete space.
If we wanted to borrow a word used in abstract algebra, we might use the term “Ideally Convergent.”