Why is zero the only infinitesimal real number? I am currently  reading Elementary Calculus: An Infinitesimal Approach by H. Jerome  Keisler and was wondering if someone could help me with an aspect treated in the book. 
On page 24 he says a number $\varepsilon$ is said to be infinitely small or infinitesimal if $$-a< \varepsilon < a$$ for every positive real number $a$. He then says the only real number that is infinitesimal is zero. 
I really don't get that. What I understand is that in order for a number to be considered infinitely small it has to be bigger then $-a$ and smaller then $a$. Well if I take $a$ to be $-2$ that means that $-1$ would be infinitesimal since it is bigger than $-2$ but smaller then $2$. So then how can zero be the only real number that satisfies that condition?
 A: Your example of taking $a$ to be $2$ and concluding that $1$ is infinitesimal since it is between $-2$ and $2$ is not a good example. 
The reason for this is that the definition of an infinitesimal $\varepsilon$ is that $-a \leq \varepsilon \leq a$ for every positive real number $a$.  You just picked some positive real number.  This has to be true for every positive real number.  That means $\varepsilon$ needs to be in $[-2, 2]$ and in $[-1, 1]$ and in $[-\frac{1}{2}, \frac{1}{2}]$ and in $[-\frac{1}{1000000}, \frac{1}{1000000}]$, and so on.  That same $\varepsilon$ has to be in all of these at the same time to be an infinitesimal.
The only real number that satisfies that it is between $-a$ and $a$ for every real $a > 0$ is $\varepsilon = 0$.  
So any number $\varepsilon$ other than $0$ that satisfies $-a \leq \varepsilon \leq a$ for every $a > 0$ real cannot itself be a real number, but there are plenty of infinitesimals that aren't real numbers.  As we discussed, $0$ is the only number that's both real and infinitesimal.
A: The point is that it is greater than $-a$ and less than $a$ for every $a$. So if you consider $e = -1$, you're correct that $-2 < -1 < 2$ but what about $-\frac{1}{2}$ and $\frac{1}{2}$? Clearly $-1$ does not lie between $-\frac{1}{2}$ and $\frac{1}{2}$. Likewise, if you had any non-zero real number $x$, $x$ does not lie between $-\frac{x}{2}$ and $\frac{x}{2}$. Thus the only real infinitesimal is $0$.
