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.

<b>Edited to add:</b> 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.