Explore BrainMass

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-?)

Solution Summary

This Solution contains over 100 words and calculations to aid you in understanding the Solution to this question.