Compute $\pi_1(\mathbb R P^n),$ for $n \geq 2.$
$\textbf {My Attempt} :$ Consider the covering quotient map $p : S^n \longrightarrow \mathbb R P^n.$ Let $\gamma$ be any loop in $\mathbb R P^n$ based at $[e].$ Consider the unique lift $\widetilde {\gamma}$ which begins at $e.$ Then either $\widetilde {\gamma}$ is a loop at $e$ or it is a path in $S^n$ from $e$ to $-e.$ In the former case $\widetilde\gamma \simeq *$ since $S^n$ is simply-connected for $n \geq 2$ and hence $\gamma \simeq *.$ In the later case $\gamma \not\simeq *$ for otherwise by homotopy lifting property $\widetilde {\gamma} \simeq *,$ a contradiction. Hence there are only two homotopy classes of loops based at $[e]$ in $\mathbb R P^n$ and hence $\pi_1(\mathbb R P^n) \cong \mathbb Z/2 \mathbb Z,$ for $n \geq 2.$
Can anybody check my attempt above and confirm whether it is fine or not? Thanks for your time.