Archimedean property concept

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.