Suppose that s is an irrational number and the sequence (r(n)) of rational numbers converges

to s: r(n) = a(n) / b(n), where a(n) and b(n) are integers, b(n)>0. Show that b(n)??. In other words, the

only way to get a very good approximation to an irrational number by a rational number is to have a

large denominator..

ii. Show that there is a constant C such that if m and n are positive integers then |m - 2 |> C

n^2. (Hint: Consider the product (m - 2 )(m + 2 ). This cannot be 0. )

iii. Show that the product (x-

( 2 + 3 ?? ? ?x - ? 2 - 3 ??(?x - ?- 2 + 3 ??(?x - ?- 2 - 3 ?? is a polynomial with integer

coefficients. In other words, when you multiply it out the coefficients turn out to be integers.

17. i. Suppose that s is an irrational number and the sequence (r(n)) of rational numbers converges to s: r(n) = a(n) / b(n), where a(n) and b(n) are integers, b(n)>0. Show that b(n) ->co. In other words, the only way to get a very good approximation to an irrational number by a rational number is to have a large denominator.. ii. Show that there is a constant C such that if m and n are positive integers then (m - V2 |> CM2. (Hint: Consider the product [min - {Slit-min + '5}. This cannot be a.) iii. Show that the product {it- (+\\))((x( )){(x(+ ))((x( nisapolynomielwith integer coefficients. In other words, when you multiply it out the coefficients turn out to be integers