Answer to Question #155671 in Calculus for Vishal

Question

If x is an arbitrary real number, show that there exists a unique n ∈ Z such that n ≤ x <n+1. (This is called the greatest integer in x and denoted by [x].)

Answer 1

suppose for the sake of contradiction that no such integer n exists. there is an integer m such that m<x,. we prove by induction that every n<=x. this is easily true by transivity for n<=m. so we take m as our base case.

Base case. if n=m, m<x so the claim is satisfied.

Inductive step. suppose that n<=x, and we will show that n+1<=x. indeed, otherwise we have n<=x<n+1, but we hav assumed thst no such integer n exists. Therefore, n+1<=x.

so we have shown that every integer n<=x. so we conclude that some such integer

n<=x<n+1 exists