Archimedean property concept I want to know what the "big deal" about the Archimedean property is. Abbott states it is an important fact about how $\Bbb Q$ fits inside $\Bbb R.$
First, I want to know if the following statements are true:
The Archimedean property states that $\Bbb N$ isn't bounded above--some natural number can be found such that it is greater than some specified real number.
The Archimedean property also states that there is some rational $\frac1n, n \in\Bbb N$ such that it is less than some specified real number.
Secondly, what do the above statements imply about the connection between $\Bbb Q$ and $\Bbb R?$ Does it imply that $\Bbb R$ fills the gaps of $\Bbb Q$ and $\Bbb N;$ is it the proof for $\Bbb R$ completing $\Bbb Q?$ 
Lastly, what insights have you obtained from the Archimedean property?
I am sorry if some of my questions are unclear.

Both statements you mention are true, and are two different ways of expressing the Archimedean property. For instance, if we have that $n \in \mathbb{N}$ and $M \in \mathbb{R}$ with $M$ positive, then $$ n >M \Leftrightarrow \frac{1}{n} < \frac{1}{M}$$
Regarding your second question, the Archimedean property is not the proof that $\mathbb{R}$ completes $\mathbb{Q}$. That fact is usually proved (though the two ideas are equivalent) from  the least-upper-bound axiom of $\mathbb{R}$. The Archimedean property of $\mathbb{N}$ is a consequence of the least-upper-bound axiom of the real numbers.
