MathLabs

Problem 2

Let a1,a2,…a_1, a_2, \dots be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer nn the numbers a1,a2,…,ana_1, a_2, \dots, a_n leave nn different remainders upon division by nn. Prove that every integer occurs exactly once in the sequence.
Step 5 of 5: Every integer occurs exactly once
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∀m∈Z, ∃! n: an=m\forall m\in\mathbb Z,\ \exists!\,n:\ a_n=m
Detailed analysis

Since the growing block {a1,…,an}\{a_1,\dots,a_n\} eventually contains every integer, every integer mm equals some term ana_n; and by the first step it equals at most one term. Hence every integer occurs exactly once in the sequence.