Existence of the decomposition, without guessing A, B and C

Second method. Justify that for every n ≥ 1 there exist integers A(n), B(n) and C(n), independent of γ, β₀ and β₁, such that f(n) = γ·A(n) + β₀·B(n) + β₁·C(n). (Question IV.2.a of the assignment.)