Proof:

Recall example (25.4) that $\mathbb{R}P^2$ arises from $S^1$ by attaching a two cell using the attaching map $f : S^1 \longrightarrow S^1$ given by $f(z) = z^2$. Since $H_1(f):\mathbb{Z}\longrightarrow \mathbb{Z}$ is given by $n\mapsto 2n$ the result immediately follows from theorems (35.4)-(35.5).