From Husemoller, page 8 (rephrased):
Theorem: Let $E, B \in \mathbf{Top}_*$, and $F \to E \to B$ be a Serre fibration. Then there is a natural group homomorphism $\partial: \pi_n(B) \to \pi_{n-1}(F)$ such that the sequence $$\ldots \to \pi_n(E) \to \pi_n(B) \ \xrightarrow{\partial}\ \pi_{n-1}(F) \to \pi_{n-1}(E) \to \ldots$$ is exact.
Exercise: apply this theorem to:
1. $\mathbb{Z} \to \mathbb{R} \to S^1$,
2. $\mathbb{Z_2} \to S^n \to \mathbb{R}P^n$,
3. $S^1 \to S^{2n+1} \to \mathbb{C}P^n$.
How to approach this? The definition via homotopy lifting is not very helpful. Husemoller mentions that it is sufficient to check each individual cell in a CW complex, but does it mean that I only have to check for $I^k,\ k = 1,\ldots,n$ [fixed bad typo]? Some hint would be nice, I'm not very familiar with CWs besides what Husemoller himself mentioned earlier (although I did make sure I understood all the proofs he gave).
PS: I asked for a lot of hints recently, it worries me. Were these questions hint-worthy?
It is equivalent to ask that maps from finite products of unit intervals into the base space lift. That is, you only need to lift maps $f:I^n\rightarrow B$ ($n$ is a non zero integer, $I=[0,1]$). This is no problem for the 2 first examples since they are covering spaces, and the total space is simply connected.
Indeed, under mild local conditions (like Hausdorff, connected and local connectedness I think) for $(X,x_0)$ a pointed space, and $p:E\rightarrow B$ a covering space, $b_0\in B$ a base point, and $e_0\in E_{b_0}\subset E$ a point in the fibre over $b_0$, and $f:X\rightarrow B$, $f$ lifts to $E$ iff $f_*\pi_1(X,x_0)\subset p_*\pi_1(E,e_0)$.
This works for the first 2 cases, because $I^n$ is simply connected, and the two fibrations are covers.