Proof:

We know that the fundamental group of $S^n$ is the trivial group and the standard quotient map $\eta: S^n \longrightarrow \mathbb{R}P^n$ is a covering projection. So from (iv) of the preceding theorem we get

$$\vert\eta^{-1}(x_0)\vert = 2 = [\pi_1(\mathbb{R}P^n, x_0) : \eta_*(\pi_1(S^n, {\tilde x}_0))]$$

From which follows that

$$\vert\pi_1(\mathbb{R}P^n, x_0)\vert = 2$$

and that completes the proof.