Question

please prove 9.6 and 9.7

The next three theorems formalize what you may have discovered in the preceding group of questions. 9.6 Theorem. Let K be a positive integer Then, among any k real num- bers, there is a pair of them whose difference is within 1/K of being an integer When we take our collection of real numbers to be multiples of an ir- rational number, then we can find good rational approximations for the irrational number. Remember how multiples of an irrational could lead to rational approximations of the irrational by finding multiples whose differ- ence is close to an integer. 9.7 Theorem. Let a be a positive irrational number and K be a positive integer. Then there exist positive integers a, b, and c with 0 sa<bsk and 0 se ka such that 17a - b - a)2 Theorem 9.6 told us that increasingly large collections of real numbers contain pairs whose differences get increasingly close to being an integer. Now you will need to understand your proof of the above theorem suf- ficiently well so that you can figure out how to make (b - a) arbitrarily large. You might consider the fact that for an irrational number a, any fixed, finite collection of multiples of a will have every difference of every pair of those multiples differing from being an integer by at least some specific non-zero amount. So taking a yet bigger collection of multiples will give you a pair whose difference is even closer to being an integer. That obser- vation might help to generalize your technique to prove Dirichlet's Rational Approximation Theorem.

Answer 1

9.6: To prove Sol,:- - To prove such type of approximatione which involves that is any large collection of real numbere, some pair of them must have a difference that is close to an integer in value. So; according to theorem; if there is any positive integerk; then among any kr humbure, there is a pair of them whore Difference is within / of being an integer So for this we used to consider a good approximation to 52, 53 & like this Suppore we consider it then its multiple to obtain good approximation to an irrational number is a valuable to understand eg:- 0h, 52, 255, -, 102.

9.7 het L be a positive irrational number & k be a positive integer; then there exist positive in a, b, c with ocasbek & oscake lies between a sm 7. Here a vay mejor challenge of rational approximation is to find fraction /b-a that between a smaller distance of the target irrational. i.e. within 1 (b-ase. One technique for finding such approximation involves noticing that in any large wllection of real numbert; some part of them must have a difference that is close to being an integer in value. egosh, 54, 254, 354, 1 1/2 - 1 e 2 But According t giver theorem; we can write above equation equality as