Question

48. For every real number x, there is a positive integer n
such that
​

Answer 1

Step-by-step explanation:

If x and y are arbitrary real numbers such that x<y, prove that there exists at least one rational number r satisfying x<r<y, and hence infinitely many. Solution: Since (y − x) > 0, by the Archimedean property, there exists a positive integer n such that n(y − x) > 1. ... Dividing by n, we get (m/n) ≤ x < (m + 1)/n < y.