Suppose that $p:E \to B$ is a weak fibration with a section $s:B \to E$ ($p \circ s=id_B$). Also here $F=p^{-1}(b_0)$.
I need to show that the sequence $0 \to \pi_n(F,s(b_0)) \xrightarrow {i_*}\pi_n(E,s(b_0)) \xrightarrow {p_*} \pi_n(B,b_0) \to 0$ is exact for every $n \geq1$.
Only thing that I don't seem to get is the exactness at $\pi_n(F,s(b_0))$. From other posts I have read that this follows from the boundary map being identically zero, since $p_*$ is surjective/onto but this is not explained any further. Could someone enlighten me how this follows?
If $p_*$ is surjective (which follows from the existence of section) then the boundary map $\partial:\pi_n(B)\to\pi_{n-1}(F)$ in the long exact sequence of homotopy groups has to be the zero map (because its kernel is the image of $p_*$). And this means that $i_*$ is injective, because its kernel is the image of $\partial$ which is trivial. So I can cut the long exact sequence and simply add $0$ before $i_*$, since we already know that it is injective.