Let a b R Suppose that a 0 Prove that there is some n N suc

Let a, b R. Suppose that a > 0. Prove that there is some n N such that b [-na, na].

Solution

use archimedean property

Let x be any real number. Then there exists a natural number n such that n>x.

n1>b/a i.e. an1>b

n2>-b/a i.e. b>-an2

where b/a and -b/a are real number, n2,n2 natural.

choose n larger of n1 and n2

an>an1>b

an2 si posiitve, an >an2 ,-an2>-an

b>-an2>-an

an>b>-an

thereofre b lies between -an and an.