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Let λ be a positive irrational real number. If n is a positive integer, choose by...

Let λ be a positive irrational real number. If n is a positive integer, choose by the Archimedean Property an integer k such that kλ ≤ n < (k + 1)λ. Let φ(n) = n − kλ. Prove that the set of all φ(n), n > 0, is dense in the interval [0, λ]. (Hint: Examine the proof of the density of the rationals in the reals.)

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