Geometrical meaning of Archimedean Property I Know the Archimedean Property of Real Numbers. 

If $x>0$ and if $y$ is an arbitrary real number, then there is a positive integer $n$ such that $nx>y$

But in Apostol's Mathematical Ananlysis, the author says, 

Geometrically, it tells us that any line segment, no matter how long, can be covered by a finite number of line segments of a given positive length, no matter how small.

I can't understand the geometrical notion of this property.
Any help?
 A: Perhaps a more accessible description of this property is in terms of the absence of infinitesimals.  An infinitesimal $\epsilon>0$ is a number so small that no matter how many times you add it to itself (provided it is a finite number of times), the resulting number $\epsilon+\epsilon+\epsilon+\ldots+\epsilon$ is still infinitesimal, meaning that it is smaller than $1$, and smaller than $1/2$, etc. The Archimedean property essentially says that the real line contains no infinitesimals. The property concerning "a finite number of line segments, etc." is a way of formalizing this without mentioning infinitesimals.
A: This simply says that, since the original line segment "no matter how long" has finite length, y, dividing it into pieces of length x, then dividing y by x gives an integer value, n, plus some fractional part. n+1 is the number of segments of length x that will cover the original length.
A: Explicitly, if $x > 0$ is arbitrary (i.e., arbitrarily small) and if $a < b$ are arbitrary real numbers (i.e., so that $y = b - a$ is arbitrarily large), there exists a positive integer $n$ such that $y < nx$.
Since $a < b = a + (b - a) = a + y < a + nx$, the (arbitrarily long) interval $[a, b]$ is contained in the finite union
$$
[a, a + nx] = \bigcup_{k=1}^{n} \bigl[a + (k-1)x, a + kx\bigl],
$$
and for $1 \leq k \leq n$, the interval $\bigl[a + (k-1)x, a + kx\bigl]$ has (arbitrarily small) length $x$.
