linear algebra – Inequality for dimensions of eigenspaces

I want to proof that for an endomorphism $f$ over vector space $V$ with eigenvalue $lambda$ and
$V = U oplus V/U$ with $f(U) subseteq U$ the following holds:
$$
dim E(f,lambda) leq dim E(f_U,lambda) + dim E(f_{V/U}, lambda)
$$

I managed to proof that the inequality holds for the algebraic multiplicity of $lambda$ via splitting the characteristic polynomial of $f$. However I can’t find a way to connect that to the geometric multiplicity. I would appreciate any hint.