Show that if r is an irrational number, there is a unique integer n such that the distance between r and n is less than I/2.

Solution:Step-1: In this problem we need to show that if r is an irrational number , there is a unique integer n such that the distance between r and n is less than . We know that irrational number is not an integer. Given that , r is an irrational number. Therefore , it must lie between two integers , So there exist an integer n such that n < r < n+1. Given r is an irrational number . So , . Now , there are two cases .Step-2: Case(1) : Therefore , the distance between r and n is less than .Step-3: Case(2) : , since...