Induced image of the fundamental group of a covering space I have been reading Hatcher´s Algebraic Topology, and he wants to prove that if we have a covering space $(E,p)$, with $p(e)=x_0$ then $p_*(\pi_1(E,e)$) consists of the homotopy classes of loops in $X$ starting at $x_0$ such that their lifts are loops in $E$ starting at $e$.To do this in the proof he says that a loop representing an element of the image $p_*$ is homotopic to a loop having such a lift , and intuitively it seems right but i cant seem to see why this is true theoretically , so any help is apreciated, Thanks.

This is basically just by definition.
Let $\gamma$ be a loop on $x_0$ that represents $p_*([\vartheta])\,\in\pi_1(X,x_0) $ with $[\vartheta] \in\pi_1(E,e)$.
This means $[\gamma] =p_*([\vartheta]) =[p\circ\vartheta]$, that is, $\gamma$ is homotopic to $p\circ\vartheta$, which lifted to $e$ obviously gives $\vartheta$. 
