How to resolve grading dispute between TA and Lecturer?

The nature of the dispute makes this problem difficult.

As a mathematics (BS) and computer science (MS, PhD) student I have done numerous exercises that required proof of the existence of a natural number N such that for all n>N some inequality is true. In addition to limits in mathematics, they show up in computational complexity analysis of algorithms.

Every time I have done one of those exercises I have picked a value of N that made the proof as simple and clear as I could. Often, I was aware of a smaller value of N that would have required a longer proof. I have never been marked down for picking an unnecessarily large value of N.

Any finite value N, no matter how large, such that the inequality is provably true for all n>N is equally good. That is an important aspect of these definitions, something the students should understand and apply.

If smallness of N were going to be a grading factor, despite its irrelevance, it should have been announced in advance.

That said, it would have been better for the OP to discuss the matter privately with the professor, and perhaps with more senior professors. The OP should not encourage protests directly, but should state the professor's decision and recommend that follow-ups be forwarded directly to the professor or offer to forward them on the students' behalf.

Mathematics allows for objective truth. If students answer a question correctly then they deserve full credit. I do not think it is wrong for you to advocate for your students or for you to encourage them to advocate for themselves.

Mathematically, you are clearly right. Any reasonable person should agree with you. The problem asked to prove that a limit holds, they proved it, period. "Find the optimal N for a given epsilon" has nothing to do with the question asked[0]. Since your professor doesn't agree with you, it makes me suspect he's not a reasonable person.

Having said that, it is still annoying for him if you "go against him" by telling the students to appeal the grade (appeal which they would win, if it is done honestly). Have you ever discussed this with him prior to you discussing it with the students? What did he say?

So why don't you propose to your professor a compromise? Ask him to change the question from "prove the limit" to "find the optimal N such that this inequality holds". Or "Once you prove the limit, give an estimate of smallest N such that the error is lower than epsilon. "

You can sort of add some context to the question to make it more sensible, for example by saying that f(n) is the percentage of criminals arrested as a function of the amount of money spent, and you want to get to a certain percentage.

In short, if he wants to ask a question about the optimality of N, make him ask that question, not an unrelated one.

[0] Personally, I would argue that it is actually harmful. Understanding that any finite intervals can be ignored and that we should focus on what happens for N arbitrarily large is a crucial point to understand convergence and limit at infinity. This obsession on the exact optimal N is harmful, because it gives the impression that it matters; it would be more beneficial to instead show how a complicated inequality, for example, can be simplified by simply considering N incredibly and unreasonably big. It doesn't matter, because we are only concerned about what happens at infinity.