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$\{a_n\}$ is a sequence of integers such that there are infinitely many positive as well as infinitely many negative integers in the sequence. Let for each $r$, the remainders of $a_1, \dots , a_r\bmod r$ are distinct. Then I need to show that every integer occurs exactly once in the sequence. I don't know in which direction to proceed. Any help or hints will be appreciated.

user26857
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Amanda Rowling
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  • I don't know the answer, but ... Where is this question coming from? - What is its background? (Some research? A math competition of some sort?) Just curious... –  Oct 25 '20 at 08:32
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    Hint: show $,a_1,\ldots, a_N,$ forms a consecutive sequence of $N$ integers (when reordered), by noting that if any elements differed by $,\color{#c00}K\ge N,$ then two remainders would be equal in $,a_1,\ldots, a_{\color{#c00}K}\ \ $ – Bill Dubuque Oct 25 '20 at 09:17
  • And, of course $\Bbb Z,$ is the only consecutive sequence of integers that has has no lower or upper bound. – Bill Dubuque Oct 25 '20 at 09:29

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