# Probability of Approximating Real Numbers with Rationals

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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