Question

Let $A=\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4},...\}.$ Prove that for every positive integer $n\ge 3$ the set A contain an infinite non-constant arithmetic sequence.

Answer 1

For $n=3$ we can write $\frac{1}{6},\frac{1}{3},\frac{1}{2}.$ as $\frac{1}{6},\frac{2}{3},\frac{3}{2}.$
Considering the arithmetic sequence $\frac{1}{n!},\frac{2}{n!},...,\frac{n}{n!}.$
When we write the fractions in their lowest terms, we see that all belong to A. For the second part, just observe that every non-constant, infinite arithmetic pro-gression is necessarily an unbounded sequence. Since A is bounded, it cannot contain such a sequence.