Reference: Lemma 5.3 from "Groups That Affect Circles" by Etiyenne Ghys.

To let $ f: mathbb {Z} rightarrow mathbb {R} $ be a quasi-homomorphism, i. $ | f (a + b) -f (a) -f (b) | leq D $ $ forall $ $ a $ and $ b $ in the $ mathbb {Z} $ ($ mathbb {R} $ and $ mathbb {Z} $ are considered here as additive groups and so you see the plus sign). I have to prove that there is a unique number $ tau $ $ epsilon $ $ mathbb {R} $ so that $ f (n) -n tau $ is limited.

I have shown the part of the uniqueness separately and have found limits that work in the following limiting cases:

(i) If $ n $ is positive, I've limited it up through $ f (0) $ With $ tau = f (1) + D $,

(ii) if $ n $ is positive, I limited it down through $ f (0) $ With $ tau = f (1) -D $,

(iii) if $ n $ is negative, I limited it up by $ f (0) $ With $ tau = f (-1) + D $,

(iv) If $ n $ is negative, I limited it down through $ f (0) $ With $ tau = f (-1) -D $, I understand that I have always used $ f (0) $ tying them, but that's just what I can see since I split $ n $ as $ n $ time the generator $ 1 $ $ epsilon $ $ mathbb {Z} $, Please help me to bind this with only one of a kind $ tau $,