Proving that an integer, $x = q+s$ exists using the well ordering principle From Serge Lang's Undergraduate Algebra
Let $x$ be a real number. Prove that there exists an integer $q$ and a real number $s$
which is greater than or equal to zero and strictly less than $1$, such that $x = q+s,$ and that $q$ and $s$ are uniquely determined.
I understand that the basic implication of the well-ordering principle is that if you can prove there is a least element in a set, then the set is nonempty. However, in this scenario
I'm struggling to see how it could be applied. Only $s$ is bounded, whereas $q$ could be anything. Should I even be using the well ordering principle?
 A: *

*Prove there exists an integer less than or equal to $x,$ maybe using the Archimedean property of $R$.

*Prove there is a greatest integer less than or equal to $x,$ maybe with proof by contradiction.

*Show that $x+1$ is the least integer greater than $x.$

*Show uniqueness of $s$ and $q$ by assuming that $s'$ and $q'$ also exist and then arrive at a contradiction.

Or something like that.
I'm not sure you need $3.$ But those some ideas. I don't have your book, so the exact way you answer your questions depends on the details in your book up until this question...
A: Maybe you misunderstood the author's words in the well-ordering of natural numbers. What the author says is that (in page 2):

Every non-empty set of integers $\geq$ 0 has a least element.
(This means: If $S$ is a non-empty set of integers $\geq$ 0, then there exists an integer $n \in S$ such that $n \leq x$ for all $x \in S$.)

Your words seems taking it in the wrong direction:

if you can prove there is a least element in a set, then the set is nonempty.

It should be "if the set is nonempty,  there is a least element in it".
Then for this exercise, I have to use a fact(I don't know if the author relies on it or not):

(i) For any real number there is some interger $\geq 0$ greater than this real number.
(ii) Especially, there is some integer $N \geq 0$ such that $N > x$

just take the set to be:
$$
S = \{m \in \Bbb{Z} |  m \geq N - x \}
$$
. From (i) $S$ is nonempty, then apply the well-ordering of natural numbers to conclude that $S$ has a least element $n$. And then take $q = N -n$ and $s = x - q$
