### 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