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!