I want to show that $$H^i(\mathbb{R}P^n; \mathbb{Z_2}) = \begin{cases} \mathbb{Z_2} \quad 0 \leq i \leq n \\~\\ 0 \quad \text{otherwise} \end{cases}$$ Using the fact that $S^n$ is the double cover of $\mathbb{R}P^n$ and there is a long exact sequence
$$ \cdots \rightarrow H^i(\mathbb{R}P^n;\mathbb{Z_2}) \rightarrow H^i(S^n;\mathbb{Z_2}) \rightarrow H^i(\mathbb{R}P^n;\mathbb{Z_2}) \rightarrow H^{i+1}(\mathbb{R}P^n;\mathbb{Z_2}) \rightarrow \cdots $$
I know that $$H^i(S^n;\mathbb{Z_2}) = \begin{cases} \mathbb{Z_2} \quad i = 0,n \\~\\ 0 \quad \text{otherwise} \end{cases}$$ but don't know how to continue. I'd appreciate any help.