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.

Let's look at some concrete consequences for ordinary calculus. Consider the following limit:
$$\lim_{n\to\infty}\frac1n=0$$
I bet you want that equation to be true, right? Well, it requires the Archimedean property! Otherwise, the sequence doesn't converge.* That would be really inconvenient. Similarly, definitions like
$$\exp(x)=\sum_{n=0}^\infty\frac{x^n}{n!}$$
rely on the Archimedean property, since that sum doesn't converge otherwise.*
*Statements about non-convergence refer to the order topology of a non-Archimedean ordered field. I make no claims about non-ordered fields, such as the $p$-adic numbers.
