Let $x\in \mathbb{R}$ , Using the Well-Ordering Property of $\mathbb{N}$ and the Archimedean Property of $\mathbb{R}$, show that there exist a unique $a \in \mathbb{Z}$ such that $a \leq x < a+1$
My approach so far:
Suppose $x$ greater than some Integer $a$.
$x\geq a$
By Archimedean Property of $\mathbb{R}$, there exist $n_{x} \in \mathbb{N}, x<n_{x}$
Combining those 2 inequality I have $a\leq x < n_{x}$.
But I do not know how to proceed from here, any help or insights is deeply appreciated.
Thank you for reading my post.
Best Answer
I'd break this up into 3 cases
Case 1: $x=0$
This case is really simple so I'll leave it for you to do.
Case 2: $x>0$
Let $S=\{n\in \mathbb{N}:n>x-1\}$, which exists by the Archimedean postulate. By the well-ordering property $S$ contains a least element $a$. Obviously $a>x-1$ Now assume by way of contradiction that $a-1>x-1$. Thus $a-1\in S$ contradicting $a$ being the least element of the set. So then $a-1\leq x-1$. We can therefore conclude that there exists a $a$ such that $a \leq x <a+1$ (by using our properties of the order axioms.)
Case 3: $x<0$
This argument is very similar to the one used in case 2. Note that $\mathbb{Z^-}=\{-n:n \in \mathbb{N}\}$. Thus we can construct any subset in the negative integers by mapping the corresponding subset in the positive integers under $f:\mathbb{N} \to \mathbb{Z^-}$ where $f(n)=-n$. We can see that $f$ is bijective. Finally remember that for positive $x,y\in \mathbb{Z}$ if $x<y$ then $-x>-y$. With the above information you can now use the well-ordering property of the positive integers to see that every subset of the negative integers has a greatest element.