Show that if B is simply-connected, then p is a homeomorphism.

Solution 1:

Your proof works fine, but you should say that $p\psi$ is homotopic rel endpoints or path homotopic to a constant path and, since a path homotopy lifts to a path homotopy, $ψ$ is path homotopic to a constant map, too, which implies $a=b$.
With little more effort, we can show that if a loop $\phi$ at $b_0$ in $B$ is in the image $p_*(\pi_1(E,e_0))$ (which is a subgroup of $\pi_1(B,b_0)$), then $ϕ$ lifts to a loop in $E$. For if $[ϕ]\in p_*(\pi_1(E,e_0))$, then there is a loop $\lambda$ at $e_0$ such that $pλ$ is path homotopic to $ϕ$, and this homotopy lifts to a path homotopy $λ\simeqψ$, where $ψ$ is the lift of $ϕ$ at $e_0$, which is thus a loop.
In particular this implies that a null-homotopic $ϕ=pψ$ lifts to a loop $ψ$ since $[ϕ]=0$ is always in $p_*(\pi_1(E,e_0))$. Therefore $ψ(0)=ψ(1)$.