Misunderstanding of $\epsilon$-neighborhood I am given the following definition of an $\epsilon$-neighborhood:
Given a real number $a\in \mathbb{R}$ and a positive number $\epsilon>0$, the set
$$\{x \in \mathbb{R}: |x-a|<\epsilon \}$$
is called the $\epsilon$-neighborhood of a.
The book says that the $\epsilon$-neighborhood of a consists of all those points whose distance from a is less than $\epsilon$. I don't understand how $|x-a|<\epsilon$ says that though. To my understanding $|x-a|<\epsilon$ says that
$$-\epsilon<x-a<\epsilon$$
$$-x-\epsilon<-a<\epsilon-x$$
$$x+\epsilon>a>x-\epsilon$$
$$x-\epsilon<a<x+\epsilon$$
Could someone please reconcile my understanding? 
Thanks!

You might want to write it this way: $$|x-a|<\epsilon \iff -\epsilon < x-a < \epsilon \iff a-\epsilon < x < a+\epsilon$$And the last expression means that $x \in \left]a-\epsilon,a+\epsilon\right[$. The neighbourhood is centered in $a$, not $x$. The moral of the history is that $|x-a|<\epsilon$ if and only if $x \in \left]a-\epsilon,a+\epsilon\right[$. So: $$\{x \in \Bbb R \mid |x-a|<\epsilon\} = \left]a-\epsilon,a+\epsilon\right[. $$
