When should an equation be numbered when writing a paper? This is a soft question on writing math for research or publication purposes. 
When do we number an equation? Or, when should we not number an equation?
Obviously, we number equations when we wish to refer back to them at some later point. But I have observed in textbooks and many publications that the numbering seems to be inconsistent. 
For example: Can anyone explain why the top two equations are not numbered, but the bottom ones are? https://arxiv.org/pdf/1201.6656.pdf

Is there a stylistic guide as to when a equation should or should not be numbered?
 A: The reason I think is because the first two equations serve as "steps" for the third one. And so it is not as important how they got the equation, rather then the equation itself.
It's like: find $z$ such that $z = x + y$ where $x=5, y= 6$.
$$x=5$$
$$y = 6$$
$$z = x + y = 11\text{               (1.2)}$$
$z$ is the most important here, not how we got it, and so in your case, the bound is the most important, rather then the thinking that went behind it. This is my guess. I'm not sure tbh.
A: If I am submitting a paper, which will undergo peer-review, I always number all of the equations. This will make the reviewing process much easier.
A: The dominant philosophy in most mathematical writing is to only number equations that are referred back to within the paper/article/note/book, or to number particularly important equations.
In general, there are three conflicting trains of thought with respect to numbering of equations in a paper.


*

*Fisher's Rule: Number every equation, every time.

*Occam's Rule: Number only those equations which are referred back to.

*Fisher-Occam Rule: Number those equations which might be referred back to.


These rules were the subject of the article Writing in the Age of LaTeX appearing in the 1995 Notices of the AMS. But even there, some of the reasons for Fisher's Rule haven't aged particularly well.
A: From a LaTeX and physics point of view -- and this depends on the style imposed by the journal or other authority:


*

*Number (almost) all display maths.
The "almost" refers to the occasional separate equation often preceding a numbered equation (so you can refer to such steps as "in the derivation of Eq. 1.1").

*Use display maths for anything you want to refer to, plus anything that needs it for clarity.

*Use inline maths when display isn't really needed (this is a common journal requirement in physics).


I know there are mathematics fields in which this will work nicely; I know there are others in which it won't.
A: I try to do what the other answerers say to do: only number those equations I  refer to.  But on looking at the OP's sample page, another reason to number equations occurs to me: other future writers might want to refer to some juicy equation of mine, and it would do them a service to set things up so they could write "According to Kimchilover's equation (17), blah blah".  
Added, 18 July:
For example, in the body of the paper I say,

This and that.  Clearly $$A=B$$ and so the set $S$ is bounded, proving the theorem. 

The equation I don't number is $A=B$, which is referred to only by the sentence it occurs in.  By the standard rule it gets no equation number. If I had foreknowledge of how important my work will be, I might say something like this  in the introduction of the paper, where the problem and methods are described:

Theorem 2 then follows from the application of our Lemma 1, the Krein-Milman theorem, and the simple-looking equation (17).

And then give my faux-humble formula its number (17).
A: You should only number equations that you are going to refer back. Most people are not very consistent. But now there are  LaTex packages that do exactly that.
See this post. 
A: I think that this was best expressed by physicist David Mermin in his column What's Wrong with These Equations (Physics Today, 1989):

[...]Fisher's rule is for the benefit not of the author, but the
  reader.
For although you, dear author, may have no need to refer in your text
  to the equations you therefore left unnumbered, it is presumptuous to
  assume the same disposition in your readers. And though you may well
  have acquired the solipsistic habit of writing under the assumption
  that you will have no readers at all, you are wrong. There is always
  the referee. The referee may desire to make reference to equations
  that you did not. Beyond that, should fortune smile upon you and
  others actually have occasion to mention your analysis in papers of
  their own, they will not think the better of you for forcing them into
  such locutions as "the second equation after (3.21)" or "the third
  unnumbered equation from the top in the left-hand column on p.2485."
  Even should you solipsistically choose to publish in a journal both
  unrefereed and unread, you might subsequently desire (just for the
  record) to publish an erratum, the graceful flow of which could only
  be ensured if you had adhered to Fisher's rule in your original
  manuscript.

