Generalized $\boldsymbol\lambda$-eigenspace

The generalized $\lambda-\text{eigenspace}$ is defined by: $V^f_{(\lambda)}=\bigl\lbrace v\in V\mid\exists j\,\text{ such that }\,(f-\lambda)^jv=0 \bigr\rbrace$. Suppose that $V$ is a vector space over the field $k$ and $f,g\in \operatorname{End}_k(V)$ satisfy $f\circ g=g\circ f$. Show that $g(V^f_{(\lambda)})\subseteq V^f_{(\lambda)}$.