Lemma 10.58.5. Suppose that f: n \mapsto f(n) \in A is defined for all n sufficiently large and suppose that n \mapsto f(n) - f(n-1) is a numerical polynomial. Then f is a numerical polynomial.
Proof. Let f(n) - f(n-1) = \sum \nolimits _{i = 0}^ r \binom {n}{i} a_ i for all n \gg 0. Set g(n) = f(n) - \sum \nolimits _{i = 0}^ r \binom {n + 1}{i + 1} a_ i. Then g(n) - g(n-1) = 0 for all n \gg 0. Hence g is eventually constant, say equal to a_{-1}. We leave it to the reader to show that a_{-1} + \sum \nolimits _{i = 0}^ r \binom {n + 1}{i + 1} a_ i has the required shape (see remark above the lemma). \square
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