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Consider the harmonic sequence

$$1, \frac{1}{2}, \frac{1}{3}, \cdots, \frac{1}{n}, \cdots$$

Prove that if we pick dinstinct terms of the above sequence we can construct an arithmetic progression sequence of as large (finite) length as we want.

Just observe that $\displaystyle{\frac{1}{n!}, \frac{2}{n!}, \ldots, \frac{n}{n!}}$ are all distinct terms of the sequence.