I'm writing my first paper and my last (main) result has a very nice intuitive reasoning, but my rigorous proof has turned into an ugly 4 page epsilon argument. The paper is on a topic in number theory (in particular showing the upper density of a certain set of integers). I have currently chosen the following plan of attack:
For every $\epsilon > 0$ I show there exists $N$ large enough s.t. 6 different quantities are all $< \frac{\epsilon}{6}$. Currently I have the bounds on the 6 different quantities as a 6 part lemma, where I find $N_1,N_2,…N_6$ s.t. for $n > N_i$ the i'th quantity is $< \frac{\epsilon}{6}$. I'm not concerned with how large my $N$ is, just that there is a sufficiently large one.
I'm quite unhappy with how my argument reads. The insight as to whats going on is completely lost in the equations. I tried compensating by giving a short paragraph explaining the reasoning behind the equations, but the reader will still have to toil through 4 pages of equations.
I was hoping to see some links to well written papers that have to deal with a similar situation that I'm in, preferably in the fields of number theory or combinatorics. I have entertained the idea of switching to big-O notation for everything and avoid the epsilon argument all together, but not sure how I'd be able to make that work. Any advice would be greatly appreciated MO!
Best Answer
Proposition 1: For sufficiently large N, the following six quantities can be made arbitrarily small:
a) b) c) d) e) f)
Proof: The proof, which is something of a technical distraction, is deferred to the end of this section.
Theorem: [Insert great theorem here.]
Proof: [Clear intuitive proof, invoking Proposition 1.]
It remains to prove Proposition 1.
Proof of Proposition 1: [Insert long boring computations here.]
Or alternatively:
Theorem: [State theorem]
Proof: I claim that for sufficiently large N, the following six quantities can be made arbitrarily small: a),b),c),d),e),f).
Granting this claim, the proof proceeds as follows: [intuitive argument here].
It remains to prove the claim:
Proof of claim a):
Proof of claim b):
Etc.
Edited to add: Also: There is absolutely no need ever to write the expression $\epsilon/6$; that's for students who are proving to their instructors that they understand what's going on. In a research paper, if you prove that six quantities can all be made arbitrarily small, you can safely assert that their sum can be made arbitrarily small and count on your readers to understand why.