How to build a $K(G,1)$ space for every group $G$? I am reading the book Algebraic Topology by Allen Hatcher.
And in page 89 when he explains how to build a $K(G,1)$ space for every group $G$:
He builds a $\Delta$-complex with the $n$-simplices $\left[g_{0},\dots,g_{n}\right]
 $ of elements of $G$, and he attaches to it the $n-1$-simplices with the corresponding vertices (the elements of $G$) and calls it $EG$.
Then, he defines an action of $G$ on $EG$ by: for each $g\in G$ taking the simplex $\left[g_{0},\dots,g_{n}\right] $ linearly onto the simplex $\left[gg_{0},\dots,gg_{n}\right]$. He concludes that this action is covering space action. 
The question:
I understand that $EG$ is a cover of $EG/G$. Why is the universal cover?
Equivalent question: why is $\pi_{1}\left(EG\right)=\left\{ e\right\} $?

He argues, beginning on line 6, that $EG$ is contractible (and thus has trivial fundamental group) by a homotopy that slides every point $x \in [g_0,\dots,g_n]$ to $[e]$ along the line in $[e,g_0,\dots,g_n]$.
Edit. Ah, too slow. :)
