prove by induction that $\pi_1(\mathbb{RP}^n) = \mathbb{Z}_2$

Question

I have demonstrated by Van-Kampen that $\pi_1(\mathbb{R}P^2) =\langle[ab] | [ab]^2 =1\rangle \cong \mathbb{Z}_2$. Using the next connections: $\mathbb{R}P^n \cong S^n\backslash \{\{x,-x\}: |x|=1\}\cong D^n \backslash\{\{\bar x, -\bar x \}: \bar x \in \partial D^n\}$ we can observe that $\{\{\bar x, -\bar x \}: \bar x \in \partial D^n\} \cong \mathbb{R}P^{n-1} $, when by the induction assumption, $\pi_1(\mathbb{R}P^{n-1}) \cong \mathbb{Z}_2$. But I don't succeed in formalizing the induction step to show that indeed $\pi_1(\mathbb{R}P^{n})\cong \mathbb{Z}_2$.