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To my understanding, in $\mathbb{R}$, we have the following ways in which a sequence can diverge:

  1. The sequence could diverge off into $\infty$ or $-\infty$ (relevant generalization)

  2. Divergence by oscillation: A sequence that neither converges to a finite number nor diverges to either $∞$ or $−∞$ is said to oscillate or diverge by oscillation.

    2.1.An oscillating sequence with finite amplitude is called a finitely oscillating sequence. eg: $\{ (-1)^n \}$

    2.2.An oscillating sequence with infinite amplitude is called an infinitely oscillating sequence. Eg: $\{ n(-1)^n \}$

In contrast, for a general metric space, is there a way to classify the ways of divergence similar to the above? If so, what are some examples of ways which a sequence could diverge in a general metric space which can't happen in $\mathbb{R}$?

Notes:

On divergence by oscillation Source

tryst with freedom
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  • Can you describe oscillation? I find it somewhat imprecise what you mean by that. – Michael Burr Apr 17 '23 at 16:02
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    There's ways I can see divergence that don't fit either of these, though "oscillation" is fuzzy. – Randall Apr 17 '23 at 16:02
  • I added a definition from some notes I read @MichaelBurr – tryst with freedom Apr 17 '23 at 16:06
  • If one defines "divergence by oscillation" to mean "divergence of whatever types is not already classified" then I suppose it is true that one gets a complete classification, but there's no reason to think that the things classified as oscillatory in this way are like what you usually think of as oscillations. – JBL Apr 17 '23 at 16:12
  • How would you define divergence by oscillation? @JBL – tryst with freedom Apr 17 '23 at 16:14
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    Personally to me the word "oscillation" suggests that there is some periodic or structured behavior going on (like say in the sequences $(-1)^n + \frac{1}{n}$ or $\sin(n)$), but according to this definition any sequence that enumerates the rational numbers is "oscillatory". Maybe for you the word "oscillatory" suggests no preconceptions, in which case you're fine; but if (like me) you have even vague ideas about what might constitute oscillation, this definition probably includes diverging sequences that don't meet your preconceptions. – JBL Apr 17 '23 at 16:18
  • (I have never in my life written down a formal definition of "divergence by oscillation" -- I classify divergence for my calculus students as divergence to $\infty$, divergence to $-\infty$, and "diverging, but in some more complicated way than going to $\infty$ or $-\infty$".) – JBL Apr 17 '23 at 16:20
  • Also a "metric space" is way too general to classify divergences. For example if $f$ is an integrable function in $\mathbb{R}$, the sequence $f_n(x)=f(x+n)$ diverges in a different way in $L^1({\mathbb R})$. – Gribouillis Apr 17 '23 at 16:33
  • I am assuming $L^1(R)$ is lebesgue integrable functions? How do you mean diverge in different way? @Gribouillis – tryst with freedom Apr 17 '23 at 16:40
  • For a general metric space I think simplest would be to consider the set of all subsequential limits of a sequence. If a singleton, then the sequence converges, and if not, then what is relevant about the set of subsequential limits (bounded, connected, etc.) will vary depending on what is relevant for whatever it is you're studying that makes use of sequences. For spaces with various additional structures, additional refinements can be included. – Dave L. Renfro Apr 17 '23 at 16:44
  • Too late to delete from my previous comment, so I'll mention that my last sentence deals with limits of functions, not of sequences -- at least when we're dealing with functions defined on spaces with a bit more structure than the positive integers (which sequences are). – Dave L. Renfro Apr 17 '23 at 16:55
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    The notion of divergence to $\pm\infty$ requires more than merely metric space structure. Recall that this notion is equivalent to convergence to $+\infty$ or to $-\infty$ in the extended real number line. Analogously, to define divergence to a point of infinity on any metric space $M$, we need a choice of compactification of $M$. – Sangchul Lee Apr 17 '23 at 16:57
  • @Sangchul Lee: I was thinking of how to incorporate the infinite limits stuff and considered talking about compactifications, but I decided to skip it because I was going on too long anyway and it occurred to me that there might be other ways of extending the space besides a compactification in which it makes sense to extend limits to. Regarding the latter, it occurs to me now (and borrowing from your phrasing) we could additionally consider ways of defining sequence (and subsequential) limits that make use of a choice of a space in which $M$ is densely embedded in. – Dave L. Renfro Apr 17 '23 at 17:20
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    There is a trivial example of divergence that happens in a way that is not possible in $\Bbb R$: consider the metric space $\Bbb Q$, with the usual metric $d(x,y)=|x-y|$. Then there are Cauchy sequences (not oscillating not divergente to $\infty$) that don't converge (in $\Bbb Q$). Can't happen in $\Bbb R$, which is complete. – Jean-Claude Arbaut Apr 17 '23 at 20:41
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    I supsect your Eg should be ${n(-1)^n}$ – Surb Apr 17 '23 at 20:55
  • What do you mean by @Fuzzy – tryst with freedom Apr 17 '23 at 21:13

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