Assuming we are thinking of the same paper, I advise you to stop
arguing with the authors: as a grad student in mathematics there are better uses of your time. Let me guess: in your first message you
explained, very politely and clearly, that the infimum of a bounded
function f(y) defined on an open subset I of R which is expressed as an infinite union of intervals I_n is not necessarily attained as the value of
f at any y in I, and hence not necessarily on any of the subsets I_n of I?

To be honest — and it’s not my intention to be mean — if I have
the right paper, I wouldn’t even necessarily hold this situation up as a
model of “how to argue with another mathematician”, since the authors
of this paper seem to be very shaky on their knowledge of basic
mathematics.

For the sake of discussion, let’s broaden the question to what to do when you see a piece of mathematics that you are reasonably sure is not
correct. Here is what I would do (I don’t claim it’s the best…):

1) Double and triple check that what you think is wrong really is.
Depending upon the situation — e.g., before writing to a famous or eminent person — show the piece of mathematics to a friend and ask
them for their opinion in a nonleading way.

[In this case, the double and triple-checking can be done almost instantaneously. and I wouldn't bother anyone else about it.]

2) If you are still sure that there is a mistake, go ahead and send an email
to the authors if you wish to. You are not obligated to do so, but there are
situations where not doing so is creating more trouble than it is worth: e.g., you may be sparing an editor and a referee the time and trouble
of looking at this piece, and if the error is a very subtle one, there is
the risk that it will be published, which will be unfortunate for many people.

3) In your first email you should be maximally polite — for one thing, there is the chance that you are completely wrong, and here your politeness will — one hopes– save you from alienating the wrong person.
You don’t have to use the word “error” or “incorrect”; by a more subtle
use of language you can communicate the exact same content. But you should point out very specifically where you think the problem (or “issue
with your understanding”) lies. The author’s (s’) first reaction is likely to be that the problem is an expository one, i.e., that they didn’t express clearly enough what they had in mind. But then they will probably go over
their argument very carefully, and either find the mistake or produce a new version.

4) If their response acknowledges the mistake and tries to correct it in a way which still seems to be inadequate, then you may be taken back to square one. (That is why it is nice to give counterexamples to a theorem
itself rather than the proof, if at all possible; it’s much more convincing that the argument is faulty if the theorem is false!) However, if you feel
that the authors are just patching up things in a superficial way then after
a few iterations you should demur, saying that you look forward to reading a totally new version of their paper once they have had the time to work
things out properly.

5) If the authors don’t acknowledge the error you pointed out — either through willful dismissal or just a lack of understanding — then there’s very little that you can gain by communicating with them further about it. Some manuscripts are predicated on a sort of passive-aggressive ignorance of mainstream mathematics — people who want to prove famous conjectures or reprove deep theorems in a few pages by totally
elementary methods are not only probably wrong, but are likely not to
respond to criticism in the way you would want. I guess you know that the
“General Mathematics” subject on the arxiv is a de facto repository for “crankish” papers (though there are some legitimate papers there as
well). The fact that someone’s purported proof of the Riemann hypothesis
appears there rather than cross-listed under analysis, number theory, and
spectral theory (or whatever) is a sign that one of the arxiv moderators
thought their paper was probably not worth troubling the analysts etc. about.

6) Some people respond to criticism by simply uploading new versions. Each version responds to the glaring error that someone pointed out in the
previous version but makes a new glaring error (or tries to reclothe the
old one). It seems that some people are happy to go to bed at night
thinking, for the 59th time, that they have proven a fantastic theorem, and
the fact that they have been wrong 58 times before does not limit their
enthusiasm.

In summary: it is not your job to fight — or even politely but repeatedly
disagree — with someone about the correctness of their work: that is the
job of the authors. The real question is: who besides you is going to read
this work, and what are they going to think? If you think that the work is
important enough to be widely read and that some of the readers will be misled by it, there are other avenues available to you, from contacting
the editor at the appropriate journal or publishing a piece yourself
responding to the errors. I have had the occasion to do the first myself (eventually resulting in an “erratum” which was really a retraction), and I am glad not to have done the second, but sometimes I have thankfully read negative reviews of popular books (e.g. David Foster Wallace’s book about infinity). But if it’s just some paper posted out there that other well-qualified people will view in a similar way…often it’s best just to let it go.

g.amazeen@gmail.com