Chapter 1.4, Problem 9E

(a) Show that every nonempty subset of $\mathbb{Z}$ that is bounded above has a largest element.

(b) If $x\notin \mathbb{Z}$, show there is exactly one $n\in \mathbb{Z}$ such that $n<x<n+1$.

(c) If $x-y>1$, show there is at least one $n\in \mathbb{Z}$ such that $y<n<x$.

(d) If $y<x$, show there is a rational number $z$ such that $y<z<x$.