= Solution
As usual for <Complex projective space> and <Real projective space>, <homogeneous coordinates> are represented by nonzero vectors. Componentwise <complex conjugation> descends to the <diffeomorphism>
$$
\tau:\mathbb{CP}^n\longrightarrow\mathbb{CP}^n,\qquad [z]\longmapsto[\overline z].
$$
It is well defined because $\overline{\lambda z}=\overline\lambda\,\overline z$, and $\tau^2=I$. Every standard <manifold chart> is invariant under $\tau$, with $\varphi_i\circ\tau=\sigma\circ\varphi_i$. Part (d) and the gluing of the <Fubini-Study form> therefore imply $\tau^*\omega_{FS}=-\omega_{FS}$.
A point $[z]$ in the <fixed-point set> satisfies $\overline z=\lambda z$. Applying <complex conjugation> again gives $|\lambda|=1$. Choose $a\ne0$ with $a/\overline a=\lambda$; then $\overline{az}=az$, so $[z]$ has a real representative. Conversely every real representative determines a point in the <fixed-point set>. Hence $\operatorname{Fix}(\tau)=\mathbb{RP}^n$, and part (c) gives
$$
\boxed{\mathbb{RP}^n\subset(\mathbb{CP}^n,\omega_{FS})\text{ is a Lagrangian submanifold}.}
$$
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