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    Probability of Approximating Real Numbers with Rationals

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    We want to show that a real number, chosen at random from the interval [0,1], can be approximated to within 1/(n²) by rational numbers of the form k/n. To do this we consider for ?>0 the event
    A(n)= {|X - k/n| < n^(-2-?) for one k = 0,1,...,n}
    Determine the probability that only finitely many of the A(n) occur.

    Here n^(-2-?) means n to the power of (-2-?).

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    Solution Summary

    This solution contains over 100 words and calculations to aid you in understanding the probability of approximately real numbers with rationals.