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.

Your statements, strictly interpreted,
are not true.
You need to change the order of quantifiers.
You say
"The Archimedean property states that $\Bbb{N}$ isn't bounded above--some natural number can be found such that it is greater than any real number."
There is no natural number "such that
it is greater than any real number."
Rather,
for any positive real number $x$,
there is a natural number $n$ 
such that $n > x$.
Similarly,
you say
"The Archimedean property also states that there is some rational $1/n,n∈ℕ$ such that it is less than any real number."
Rather,
for any positive real number $x$,
there is a positive integer $n$
such that
$1/n < x$.
Be careful out there.
