MathDB

The set

S=\{ \frac{1}{n} \; \vert \; n \in \mathbb{N} \}

contains arithmetic progressions of various lengths. For instance,

\frac{1}{20}

\frac{1}{8}

\frac{1}{5}

is such a progression of length

3

and common difference

\frac{3}{40}

. Moreover, this is a maximal progression in

S

since it cannot be extended to the left or the right within

S

(

\frac{11}{40}

and

\frac{-1}{40}

not being members of

S

). Prove that for all

n \in \mathbb{N}

, there exists a maximal arithmetic progression of length

n

S

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