Here is how the principle is laid out in the text:
Given real numbers $a$ and $b$, with $a \gt 0$, there is an integer $n \in \mathbb{N}$ such that $b \lt na$
First off what is important about finding an $n$ that satisfies $b \lt na$
It talks about the strategy behind the proof and goes on to say that any non-empty subset of integers that is bounded above has "a largest integer". If $k_0$ is the largest integer that satisfies $k_0a \leq b$, then $n=(k_0+1)$ must satisfy $na \gt b$. In order to justify this application of the Completeness axiom, we have two details. 1) Is the set $E:= \{k \space \in \mathbb{N} \space : \space ka \leq b\}$ bounded above and 2) Is $E$ non-empty? This answer depends on whether $b \lt a$ or not.
What I don't understand is why is $k_0$ introduced? and why does it matter if $k_0a \leq b$? Where did the idea of $k_0a \leq b$ come from and why is it important? Why does $E$ being non-empty depend on whether $b \lt a$ or not? How does it follow that n = $k_0 + 1$ must satisfy $b \lt na$?
Then the proof is laid out:
If $b \lt a$ set $n=1$. If. If $a \leq b$, consider the set $E:= \{k \space \in \mathbb{N} \space : \space ka \leq b\}$. $E$ is nonempty since $1 \in E$. Let $k \in E$. Since $a \gt 0$ it follows from the first multiplicative property that $k \leq \frac{b}{a}$. This proves $E$ is bounded by $\frac{b}{a}$. Thus by the completeness axiom $E$ has a finite supremum $s$ that belongs to $E$. set $n= s+1$. Then $n \in \mathbb{N}$, n cannot belong to $E$ thus $ na \gt b$.
Whats the significance of if $b \lt a$ then set $n=1$? I thought we were dealing with the set $E$. How is $E$ considered non-empty when $1 \in E$, by that I mean where was the $1$ pulled from? Just the fact that we are dealing with the set $\mathbb{N}$? I'm confused. What justifies setting $n = k_0 + 1$?
Best Answer
Nothing is important about finding the $n$. What's important is to understand that is you have real positive number $a$, no matter how small, and another positive real number $b$, no matter how big, you will be able to add up some number of the teeny-tiny $a$ and end up something bigger than the huge ginormous gargantuan $b$.
This is important. It means: The integers are not bounded. It means no number so large that in can not be surpassed by adding a smaller value enough times. It means by corrolary that no positive number is so small we can't find a smaller positive number. And it means, also as a corollary that two numbers, no matter how close they are together will always have a number between them.
Now these observations may seem obvious but that are not givens. They have to be proven.
We need to rule out that we can't just keep adding $a$s together and never get above $b$. If the set of all integers $k$ where $ak \le b$ is bounded above then it must have a greatest element, $k_0$. If it has a greatest element then we are done; we just take the next element $k_0 +1$ it is not in the set of all such integers. So $a(k_0 + 1) \not \le b$ so $a(k_0 + 1) > b$.
So we would be done with our proof... IF we know the set was bounded above.
If $b < a$ then $b < 1*a$ and for any $n \in \mathbb N$ we have $n \ge 1$ so $an > b$ and there are no natural numbers where $an \le b$. So $E$ is empty.
If $b\ge a$ then $a*1 \le b$ and $1 \in E$ and $E$ is not empty.
$k_0$ is the largest natural number in $E$. $n= k_0 + 1 > k_0$. So $n$ is larger than then largest number in $E$. So $n$ is NOT in $E$. So it is not true that $an \le b$. So $an > b$.
There are two different proofs. There is one proof where $a > b$.
There is another if $a\le b$.
The proof if $a > b$ doesn't deal with $E$ at all. (For one thing, if $a > b$ then $E$ is empty.) For another if $a > b$ the proof is trivial.
If $a > b$ then $1\not \in E$. That's the problem. If $a > b$ then we can't do the proof as it is presented If $a > b$ then $E$ is empty. So we have to do another proof. The other proof is one line long.
If $a > b$ then $1*a > b$ and therefore there an $n$ where $na > b$. Because $1$ can be that $n$.
But if $a \le b$ then we have to do the "real" proof.
Let $E$ but the set of all natural numbers, $k$ so that $ak \le b$.
If $a\le b$ then $1*a \le b$ so $1 \in E$. (IF $a \le b$). So $E$ is non-empty.
Let $k \in E$. That means $ka \le b$. So $k \le \frac ba$. So $E$ is bounded above by $\frac ba$. (IF $a \le b$... if $a > b$ then $E$ is empty and none of this matters...)
Since $E$ is bounded above it has a largest element. Call the largest number ..... you know what, DON'T call it $k_0$. That is tripping you up somehow. Call $LOOKATMEIMTHEBIGGESTNUMBERINALLOFE$. $LOOKATMEIMTHEBIGGESTNUMBERINALLOFE$ is the biggest number in all of $E$.
Now let $HEYIMEVENBIGGERTHANLOOKATMEIMTHEBIGGESTNUMBERINALLOFE = LOOKATMEIMTHEBIGGESTNUMBERINALLOFE + 1$. Now $HEYIMEVENBIGGERTHANLOOKATMEIMTHEBIGGESTNUMBERINALLOFE$ is even bigger than $LOOKATMEIMTHEBIGGESTNUMBERINALLOFE$ which is the biggest number in $E$. So $HEYIMEVENBIGGERTHANLOOKATMEIMTHEBIGGESTNUMBERINALLOFE$ is even bigger than the biggest number in $E$ so it is too big to be in $E$.
So $HEYIMEVENBIGGERTHANLOOKATMEIMTHEBIGGESTNUMBERINALLOFE \not \in E$ which means $HEYIMEVENBIGGERTHANLOOKATMEIMTHEBIGGESTNUMBERINALLOFE\cdot a \not \le b$ and $HEYIMEVENBIGGERTHANLOOKATMEIMTHEBIGGESTNUMBERINALLOFE\cdot a> b$
And that's the proof.