Prove that if $x \in R,$ then there exists $n \in Z$ satisfying $x \leq n < x+1$
You will need the "well ordering property" of the positive integers- every non-empty set of positive integers contains a smallest member. (https://en.wikipedia.org/wiki/Well-ordering_principle)
Given a real number, x, Let A be the set of all positive integers strictly greater than x. By the "Archimedian property" (https://en.wikipedia.org/wiki/Archimedean_property) that set is non-empty. Let m be the smallest member of that set.
Let n= m- 1. Then n< m and since m is the smallest member of A, n is not in A. Therefore, $n\le x$. But n+ 1= m which is in A. n+ 1> x