Long exact sequence of homotopy of a Serre fibration

Statement
Suppose $$p:E \to B$$ is a Serre fibration with fiber $$F$$. Suppose $$e \in E$$, $$b = p(e)$$ and $$i:F \to E$$ is the inclusion of $$F$$ as the fiber $$p^{-1}(b)$$. Suppose $$f \in F$$. We then have the following long exact sequence of homotopy groups induced :

$$\! \dots \to \pi_n(F,f) \stackrel{\pi_n(i)}{\to} \pi_n(E) \stackrel{\pi_n(p)}{\to} \pi_n(B) \to \pi_{n-1}(F,f) \stackrel{\pi_{n-1}(i)}{\to} \dots$$

The long exact sequence ends with:

$$\! \dots \to \pi_1(B,b) \to \pi_0(F,f) \to \pi_0(E,e) \to \pi_0(B,b)$$

where the last three arrows are considered as arrows of pointed sets and exactness is interpreted in that fashion.

Note that if $$B$$ is a paracompact Hasudorff space, and $$p$$ is a fiber bundle with fiber $$F$$, then $$p$$ is also a Serre fibration with fiber $$F$$.